Preprints:
- Error estimates for vector field interpolation based on generalized matrix-valued kernels. Zhengjie Sun, Lishuo Dong, and Leevan Ling.
- Local regularity estimation through Sobolev-scale norm profile. Xiaobin Li, Leevan Ling, and Yizhong Sun.
- An adaptive Lagrangian B-spline framework for point cloud manifold evolution. Muhammad Ammad, and Leevan Ling.
- B-spline-based ALE-MFS framework for evolving domains. Muhammad Ammad, Leevan Ling, and Shu Ma.
- Sobolev algorithm for local smoothness analysis (SALSA) via sharp direct and inverse statements. Sara Avesani, Leevan Ling, Francesco Marchetti, and Tizian Wenzel.
-
Inverse inequalities for kernel-based approximation on bounded domains and Riemannian
manifolds.
Zhengjie Sun, and Leevan Ling.
Published Articles:
-
Trajectory-based RBF collocation method
via closest-point embedding for surface advection-diffusion equations.
Xiaobin Li, Leevan Ling, and Yizhong Sun
SIAM Journal on Scientific Computing. Accepted.
Abstract: We introduce a Trajectory-Based RBF Collocation (TBRBF) method for solving surface advection-diffusion equations on smooth, compact manifolds. TBRBF decouples advection and diffusion by applying a characteristic treatment with a Kansa-type RBF collocation method for diffusion PDE, which yields an operator-split characteristic (OSC) system comprising a characteristic ODE and a diffusion PDE. We rigorously prove the equivalence between the OSC system and the original surface PDE on manifolds by embedding the latter into a narrow band domain through the closest point mapping and its constant-along-normal extension. Using an intrinsic approach, we construct a time-continuous embedded PDE with push-forward operators in each chart of the atlas and establish its equivalence with the OSC system in the narrow band. Restricting the solution back to the manifold recovers the OSC system on manifolds, ensuring that the method introduces no operator splitting error. After the surface OSC system is obtained, it admits both extrinsic and intrinsic discretizations. Extensive numerical experiments confirm the robust stability and accuracy of the proposed method.
@ARTICLE{Li+LingETAL-Trajcollmethclos:26, title = {Trajectory-based {RBF} collocation method via closest-point embedding for surface advection-diffusion equations}, author = {Li, X. and Ling, L. and Sun, Y.}, journal = {SIAM J. Sci. Comput.}, year = {Accepted}, archivePrefix = {arXiv}, eprint = {2601.18186}, primaryClass = {math.NA} } -
Atlas-based meshless finite differences for elliptic partial differential equations on manifolds
represented as point clouds.
Yichen Su, and Leevan Ling
Advances in Computational Mathematics. 52:95, 2026.
Abstract: We present a systematic investigation of atlas-based radial basis function finite difference (RBFFD) methods for solving elliptic partial differential equations on smooth, compact surfaces represented as point clouds. This approach organizes point cloud data according to coordinate charts and computes differentiation weights collectively within each chart, contrasting with traditional RBFFD methods that construct independent stencils at each point. While the theoretical convergence framework has been established, practical implementation strategies and convergence behavior under different refinement regimes remain unexplored. We address these gaps by formalizing implementation strategies for both intrinsic and extrinsic formulations, including practical algorithms for atlas construction and local parameterization from point cloud data. We conduct convergence studies under two refinement scenarios: simultaneous refinement validates theoretical convergence rates, while the fixed-atlas regime reveals accuracy saturation due to geometric discretization error. We compare different local parameterization approaches and investigate weighted variants to improve method performance. Numerical experiments on various complex surfaces demonstrate that atlas-based RBFFD methods accurately solve elliptic problems without requiring explicit surface parameterizations or mesh connectivity.
@ARTICLE{Su+LingETAL-AtlasMeshless:26, title = {Atlas-based meshless finite differences for elliptic partial differential equations on manifolds represented as point clouds}, author = {Su, Y. and Ling, L.}, journal = {Adv. Comput. Math.}, volume = {52}, number = {95}, year = {2026}, doi = {10.1007/s10444-026-10369-6}, } -
Making Gaussian Kolmogorov-Arnold networks reliable and accurate.
Amir Noorizadegan, Sifan Wang, and Leevan Ling.
Neurocomputing. 703:134771, 2026.Abstract: Kolmogorov--Arnold Networks (KANs) replace fixed activations with learnable univariate edge functions whose behavior depends strongly on the chosen basis. Gaussian radial basis functions provide a simple and efficient alternative to splines, but their accuracy and stability are highly sensitive to the scale parameter $\epsilon$, which has not been studied systematically. We analyze this dependence through the geometry and conditioning of the first-layer feature matrix. Because the first layer is defined directly on the input domain, any loss of feature distinguishability introduced there propagates through the entire network. Based on this analysis, we identify the practical operating interval \[\epsilon \in \left[ \frac{1}{G - 1}, \frac{2}{G - 1} \right], \] where $G$ is the number of Gaussian centers. This interval is proposed as a stable design rule rather than a universal optimum. Extensive experiments on function approximation and physics-informed problems confirm its reliability across different collocation densities, grid resolutions, architectures, and input dimensions. We also show how the same principle supports fixed-scale selection, variable-scale models, constrained optimization of $\epsilon$, and efficient scale search using early-stage training error. These results establish scale selection as a central design principle for reliable and accurate Gaussian KANs. The code and implementation details are available at \url{https://github.com/AmirNoori68/Gaussian-KAN}.
@ARTICLE{Noorizadegan+WangETAL-MakiGausnetwreli:26, title = {Making {G}aussian {K}olmogorov-{A}rnold networks reliable and accurate}, author = {Noorizadegan, A. and Wang, S. and Ling, L.}, journal = {Neurocomputing}, volume = {703}, pages = {134771}, year = {2026}, doi = {10.1016/j.neucom.2026.134771}, } -
WLS meshless finite difference methods for time-dependent PDEs on manifolds.
Yichen Su, and Leevan Ling
Engineering Analysis with Boundary Elements. 191:106930, 2026.Abstract: The chart-based least squares meshless finite difference method for elliptic problems on manifolds implicitly approximates the inverse operator through overdetermined systems, which naturally accommodates implicit time-stepping schemes. For explicit time integration, which requires direct operator evaluation at each time step, we develop a direct operator approximation through chart-based localization. We propose a family of weighted least squares (WLS) meshless finite difference methods that includes standard RBFFD and the proven convergent least squares meshless finite difference as limiting cases. Through spectral analysis, we find that the unweighted direct formulation produces spurious eigenvalues from inconsistent approximations in overlapping chart regions, destabilizing explicit schemes. We notice that unweighted chart aggregation produces spurious eigenvalues destabilizing explicit methods, and develop a weighted formulation with spectral-informed parameter selection that mitigates these instabilities. Numerical experiments on parabolic and reaction-diffusion problems on various manifolds demonstrate spatial convergence independent of temporal discretization, showing that spectral optimized weighting facilitates stable direct operator evaluation for explicit schemes, while the framework supports pattern formation on complex geometries with both explicit and implicit methods.
@ARTICLE{Su+Ling-meshfinidiffmeth:26, title = {{WLS} meshless finite difference methods for time-dependent {PDE}s on manifolds}, author = {Su, Y. and Ling, L.}, journal = {Eng. Anal. Bound. Elem.}, volume = {191}, pages = {106930}, year = {2026}, doi = {10.1016/j.enganabound.2026.106930}, } -
Minimum-norm interpolation for unknown surface reconstruction.
Alex Shiu Lun Chu, Leevan Ling, and Ka Chun Cheung.
Computers and Mathematics with Applications. 217:220-236, 2026.Abstract: We study algorithms to estimate geometric properties of raw point cloud data through implicit surface representations. Given that any level-set function with a constant level set corresponding to the surface can be used for such estimations, numerical methods need not specify a unique target function for these domain-type interpolation problems. In this paper, we focus on kernel-based interpolation by radial basis functions (RBF) and reformulate the uniquely solvable interpolation problem into a constrained optimization model. This model minimizes some user-defined norm while enforcing all interpolation conditions. To enable nontrivial feasible solutions, we propose to enhance the trial space with 1D kernel basis functions inspired by Kolmogorov-Arnold Networks (KANs). Numerical experiments demonstrate that our proposed mixed-dimensional trial space significantly improves surface reconstruction from raw point clouds. This is particularly evident in the precise estimation of surface normals, outperforming traditional RBF trial spaces including the one for Hermite interpolation. This framework not only enhances the processing of raw point cloud data but also shows potential for further contributions to computational geometry. We demonstrate this with a point cloud processing example.
@ARTICLE{Chu+LingETAL-Miniinteunknsurf:26, title = {Minimum-norm interpolation for unknown surface reconstruction}, author = {Chu, A. S. L. and Ling, L. and Cheung, K. C.}, journal = {Comput. Math. Appl.}, volume = {217}, pages = {220-236}, year = {2026}, doi = {10.1016/j.camwa.2026.05.025}, } -
A practitioner's guide to Kolmogorov-Arnold networks.
Amir Noorizadegan, Sifan Wang, Leevan Ling, and Juan P. Dominguez-Morales.
Computer Science Review. 62:100991, 2026.Abstract: Kolmogorov-Arnold Networks (KANs), whose design is inspired—rather than dictated— by the Kolmogorov superposition theorem, have emerged as a structured alternative to MLPs. This review provides a systematic and comprehensive overview of the rapidly expanding KAN literature. The review is organized around three core themes: (i) clarifying the relationships between KANs and Kolmogorov superposition theory (KST), MLPs, and classical kernel methods; (ii) analyzing basis functions as a central design axis; and (iii) summarizing recent advances in accuracy, efficiency, regularization, and convergence. Finally, we provide a practical ``Choose-Your-KAN'' guide and outline open research challenges and future directions. The accompanying GitHub repository (https://github.com/AmirNoori68/kan-review) serves as a structured reference for ongoing KAN research.
@ARTICLE{Noorizadegan+WangETAL-guidnetw:26, title = {A practitioner's guide to {K}olmogorov-{A}rnold networks}, author = {Noorizadegan, A. and Wang, S. and Ling, L. and Dominguez-Morales, J. P.}, journal = {Comput. Sci. Rev.}, volume = {62}, pages = {100991}, year = {2026}, doi = {10.1016/j.cosrev.2026.100991}, } -
Foliation structures and global
flow dynamics of scalar hyperbolic conservation laws on manifolds: I. Geometry-compatible fluxes and
numerical
validation on sphere and torus.
Bao-Shan Wang, Alex Shiu Lun Chu, Leevan Ling, and Wai Sun Don.
Communications in Nonlinear Science and Numerical Simulation. 152:109415, 2026.Abstract: This work develops a theoretical and computational framework for scalar hyperbolic conservation laws (sHCL) posed on closed, two-dimensional (2D) manifolds. The key contribution is a geometry-compatible (GC) flux formulation based on prescribed flux directional vectors, which guarantees consistency between surface divergence and manifold geometry. This construction induces a natural foliation: the manifold decomposes into leaves along which the 2D sHCL reduces to a family of one-dimensional (1D) HCL problems. Global flow dynamics then arise as the collective evolution of these 1D leaf-wise solutions. Part I emphasizes this GC flux formulation, its foliation-based reduction, and the validation of the 2D-1D correspondence. Numerical simulations, performed using a cp-WENO scheme that combines the Closest Point Method (CPM) embedding with high-order WENO discretization, are primarily employed to substantiate the theoretical analysis. Experiments with the inviscid Burgers' equation illustrate the framework on both the sphere and torus. On the sphere, shocks and rarefaction waves evolve along circular leaves, and the longest leaf emerges as an asymptotic separatrix dividing large-scale rotational patterns. On the torus, the foliation depends critically on the flux vector: in the degenerate case, singular circular leaves act as invariant barriers separating clockwise and counterclockwise flows; in the generic case, isolated singular points anchor rotational structures and organize nonlinear wave interactions. These results confirm that nontrivial geometric features (regular and singular leaves, barriers, and separatrices) govern the global flow on manifolds. More broadly, the study establishes that 2D sHCL on manifolds can be rigorously analyzed and faithfully approximated as a collection of 1D sHCL defined along foliated leaves. Part II will extend the framework to more general fluxes and provide full details of the numerical methodology.
@ARTICLE{Wang+ChuETAL-Folistruglobflow:26, title = {Foliation structures and global flow dynamics of scalar hyperbolic conservation laws on manifolds: I. geometry-compatible fluxes and numerical validation on sphere and torus}, author = {Wang, B.-S. and Chu, A. S. L. and Ling, L. and Don, W. S.}, journal = {Commun. Nonlinear Sci. Numer. Simul.}, volume = {152}, pages = {109415}, year = {2026}, doi = {10.1016/j.cnsns.2025.109415}, } -
Energy-conserving Kansa methods
for Hamiltonian wave equations.
Xiaobin Li, Meng Chen, Zhengjie Sun, Leevan Ling, and Siqing Li.
Applied Mathematics and Computation. 510:129682, 2026.Abstract: We introduce a fast, constrained meshfree solver designed specifically to inherit energy conservation (EC) in second-order time-dependent Hamiltonian wave equations. For discretization, we adopt the Kansa method, also known as the kernel-based collocation method, combined with time-stepping. This approach ensures that the critical structural feature of energy conservation is maintained over time by embedding a quadratic constraint into the definition of the numerical solution. To address the computational challenges posed by the nonlinearity in the Hamiltonian wave equations and the EC constraint, we propose a fast iterative solver based on the Newton method with successive linearization. This novel solver significantly accelerates the computation, making the method highly effective for practical applications. Numerical comparisons with the traditional secant methods highlight the competitive performance of our scheme. These results demonstrate that our method not only conserves the energy but also offers a promising new direction for solving Hamiltonian wave equations more efficiently. While we focus on the Kansa method and corresponding convergence theories in this study, the proposed solver is based solely on linear algebra techniques and has the potential to be applied to EC constrained optimization problems arising from other PDE discretization methods.
@ARTICLE{Li+ChenETAL-EnerKansmethHami:26, author = {Li, X. and Chen, M. and Sun, Z. and Ling, L. and Li, S.}, title = {Energy-conserving {K}ansa methods for {H}amiltonian wave equations}, journal = {Appl. Math. Comput.}, volume = {510}, pages = {129682}, year = {2026}, issn = {0096-3003}, doi = {10.1016/j.amc.2025.129682}, } -
Efficient manifold
evolution algorithm using adaptive B-Spline interpolation.
Muhammad Ammad, and Leevan Ling.
Engineering Analysis with Boundary Elements. 180:106488, 2025.Abstract: This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction-diffusion systems for pattern formation.
@ARTICLE{Ammad+Ling-Effimanievolalgo:25, author = {Ammad, M. and Ling, L.}, title = {Efficient manifold evolution algorithm using adaptive {B}-Spline interpolation}, journal = {Eng. Anal. Bound. Elem.}, volume = {180}, pages = {106488}, year = {2025}, issn = {0955-7997}, doi = {10.1016/j.enganabound.2025.106488}, } -
A least-squares
generalized finite difference method for solving nonlinear reaction-diffusion systems.
Zhuochao Tang, Hui Pan, Zhuojia Fu, Meng Chen, and Leevan Ling.
Engineering Analysis with Boundary Elements. 179:106351, 2025.Abstract: Inspired by the benefits of the least-squares operation as introduced in the Kansa method (Chen and Ling, 2020), this paper introduces a least-squares framework predicated on the generalized finite difference method (GFDM). This method could be adeptly combined with a second-order semi-implicit backward differentiation formula (SBDF2), aiming to efficiently solve coupled nonlinear reaction-diffusion systems in both two-dimensional (2D) and three-dimensional (3D) spaces. The proposed Least-Squares Generalized Finite Difference Method (LS-GFDM) extends the conventional GFDM by incorporating the flexibility to collocate at arbitrary points, which brings out the advantages of both the function values approximation and the partial derivatives approximation at any given collocation point. Furthermore, this study provides an eigenvalue stability analysis of the LS-GFDM, employing the concept of variable separation. Finally, to underscore the efficacy of the LS-GFDM, several numerical examples are provided, including a benchmark test and applications to Turing pattern formations in both 2D and 3D contexts.
@ARTICLE{Tang+PanETAL-leasgenefinidiff:25, author = {Tang, Z. and Pan, H. and Fu, Z. and Chen, M. and Ling, L.}, title = {A least-squares generalized finite difference method for solving nonlinear reaction-diffusion systems}, journal = {Eng. Anal. Bound. Elem.}, volume = {179}, pages = {106351}, year = {2025}, doi = {10.1016/j.enganabound.2025.106351}, } -
Hybrid high-order shock-capturing
scheme for one-dimensional hyperbolic conservation laws on manifolds (surface PDEs) in the time-continuous
embedding framework
Wai Sun Don, Jia-Le Li, Leevan Ling, Bao-Shan Wang, and Yinghua Wang,
In: S. Chun, J. H. Jung, E. J. Park, J. Shen. (eds) Spectral and High-Order Methods for Partial Differential Equations ICOSAHOM 2023, page 239-255. Springer, Cham.2025.
Abstract: A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey's boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers' equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.
@INPROCEEDINGS{Don+LiETAL-HybrHighShocSche:25, author = {Don, W. S. and Li, J.-L. and Ling, L. and Wang, B.-S. and Wang, Y.}, editor = {Chun, S. and Jung, J.-H. and Park, E.-J. and Shen, J.}, title = {Hybrid High-Order Shock-Capturing Scheme for One-Dimensional Hyperbolic Conservation Laws on Manifolds (Surface {PDE}s) in the Time-Continuous Embedding Framework}, booktitle = {Spectral and High-Order Methods for Partial Differential Equations {ICOSAHOM} 2023}, year = {2025}, publisher = {Springer Nature Switzerland}, address = {Cham}, pages = {239--255}, doi = {10.1007/978-3-031-76988-7_12}, } -
A unified CPM framework using the
least-squares generalized finite difference method for surface PDEs.
Zhuochao Tang, Zhuojia Fu, Meng Chen, and Leevan Ling.
Engineering with Computers. 41(5), 3241-3255. 2025.Abstract: This paper introduces a novel framework, the unified closest point method (CPM) using the least-squares generalized finite difference method (LS-GFDM), to solve the surface partial differential equations (PDEs). Our approach addresses key limitations of existing embedding methods by unifying the computation of finite difference weights and interpolation into a single step, enabling efficient and reasonably accurate solutions on scattered data points. Unlike traditional CPM frameworks that rely on the finite difference method (FDM) and require interpolation to restrict solutions on the surface, potentially introducing additional errors, our LS-GFDM eliminates the need for interpolation. Furthermore, compared to the orthogonal gradient method (OrG) based on radial basis function (RBF) methods, which requires specialized data point arrangements, our method offers greater flexibility by accommodating any arbitrary point set. Higher-order schemes for LS-GFDM are easily achieved by increasing the order of the Taylor series expansion, and the method can be readily extended to other embedded methods. Numerical experiments demonstrate that the proposed framework is less sensitive to the narrow band domain size and showcases good effectiveness and robustness for solving surface PDEs.
@ARTICLE{Tang+FuETAL-unifframusinleas:25, author = {Tang, Z. and Fu, Z. and Chen, M. and Ling, L.}, title = {A unified {CPM} framework using the least-squares generalized finite difference method for surface {PDE}s}, journal = {Eng. Comput.}, volume = {41}, number = {5}, pages = {3241--3255}, year = {2025}, doi = {10.1007/s00366-025-02159-3}, } -
Learning PDEs from data on closed
surfaces with sparse optimization.
Zhengjie Sun, Leevan Ling, and Ran Zhang.
Communications in Computational Physics. 37(2): 289-314, 2025.Abstract: Discovering underlying partial differential equations (PDEs) from observational data has important implications across fields. It bridges the gap between theory and observation, enhancing our understanding of complex systems in applications. In this paper, we propose a novel approach, termed physics-informed sparse optimization (PIS), for learning surface PDEs. Our approach incorporates both $L_2$ physics-informed model loss and $L_1$ regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs. The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods. We provide practical demonstrations of the algorithms including linear and nonlinear systems. The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.
@ARTICLE{Sun+LingETAL-LearPDEsfromdata:25, author = {Sun, Z. and Ling, L. and Zhang, R.}, title = {Learning {PDE}s from data on closed surfaces with sparse optimization}, journal = {Commun. Comput. Phys.}, volume = {37}, number = {2}, pages = {289-314}, year = {2025}, } -
Structure-preserving kernel-based
methods for solving dissipative PDEs on surfaces.
Zhengjie Sun, Leevan Ling, and Meng Chen.
Journal of Scientific Computing. 102(3):70, 2025.Abstract: In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties. Furthermore, we provide numerical evidence demonstrating that the Lagrange function and the coefficients generated by a restricted kernel decay exponentially, even though a comprehensive theory has not yet been developed.
@ARTICLE{Chen+SunETAL-StruKernmethsolv:25, author = {Sun, Z. and Ling, L. and Chen, M. }, title = {Structure-preserving Kernel-based methods for solving dissipative PDEs on surfaces}, journal = {J. Sci. Comput.}, volume = {102}, number = {3}, pages = {70}, year = {2025}, doi = {10.1007/s10915-024-02774-0}, } -
A novel localized least-squares
collocation method for coupled bulk-surface problems.
Zhuochao Tang, Zhuojia Fu, Meng Chen, and Leevan Ling.
Applied Mathematics and Computation. 492:129250, 2025.Abstract: In this paper, we present a novel least-squares formulation of the Generalized Finite Difference Method (GFDM) and utilize its high-order schemes to solve the coupled bulk-surface reaction-diffusion equations. The coupled bulk-surface problems are composed of bulk equations and surface equations and coupled via some Robin-type boundary conditions. For differential operators on curved surfaces, we focus on the extrinsic definition that defines the surface operators using projection operator to tangent spaces of the surface. By utilizing localization and FD data points, the coupled model is discretized as a large sparse system using the LS-GFDM with two sets of arbitrarily distributed points. Compared with the original GFDM, the LS-GFDM brings about the advantage that it gains flexibility to use FD data points at locations different from the unknown nodal solution values. Finally, numerical demonstrations and applications of Turing pattern formations verify the effectiveness and robustness of the proposed method.
@ARTICLE{Tang+FuETAL-novelocaleascoll:25, title = {A novel localized least-squares collocation method for coupled bulk-surface problems}, author = {Tang, Z. and Fu, Z. and Chen, M. and Ling, L.}, journal = {Appl. Math. Comput.}, volume = {492}, pages = {129250}, year = {2025}, issn = {0096-3003}, doi = {10.1016/j.amc.2024.129250}, } -
Proving the stability estimates
of variational least-squares kernel-based methods.
Meng Chen, Leevan Ling, and Dongfang Yun.
Computers and Mathematics with Applications. 180:46-60, 2025.Abstract: Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.
@ARTICLE{Chen+LingETAL-Provstabestivari:25, author = {Chen, M. and Ling, L. and Yun, D.}, title = {Proving the stability estimates of variational least-squares kernel-based methods}, journal = {Comput. Math. Appl.}, volume = {180}, pages = {46-60}, year = {2025}, issn = {0898-1221}, doi = {10.1016/j.camwa.2024.12.008}, } -
Greedy trial subspace selection
in meshfree time-stepping scheme with applications in coupled bulk-surface pattern
formations.
Yichen Su, and Leevan Ling.
Mathematics and Computers in Simulation. 228:498-513, 2025.Abstract: Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.
@ARTICLE{Su+Ling-Greetriasubssele:25, author = {Su, Y. and Ling, L.}, title = {Greedy trial subspace selection in meshfree time-stepping scheme with applications in coupled bulk-surface pattern formations}, journal = {Math. Comput. Simul.}, volume = {228}, pages = {498-513}, year = {2025}, doi = {10.1016/j.matcom.2024.09.018}, } -
A high-order meshless linearly implicit
energy-preserving method for nonlinear wave equations on Riemannian manifolds.
Zhengjie Sun, and Leevan Ling.
SIAM Journal on Scientific Computing. 46(6):A3779-A3802, 2024.
Abstract: In this paper, we propose a kernel-based meshless energy-preserving method for solving nonlinear wave equations on closed, compact, and smooth Riemannian manifolds. Our method employs the scalar auxiliary variable approach to transform the nonlinear term into a quadratic form, enabling a linearly implicit scheme that reduces computational time and has good energy conservation properties. Spatial discretization is achieved through a meshless Galerkin approximation in a finite-dimensional space spanned by Lagrange basis functions constructed from positive definite functions. The method demonstrates a high order of convergence without requiring an underlying mesh. Numerical experiments validate the theoretical analysis, confirming the convergence order and energy-preserving properties of the proposed method.
@ARTICLE{Sun+Ling-HighMeshLineImpl:24, author = {Sun, Z. and Ling, L.}, title = {A High-Order Meshless Linearly Implicit Energy-Preserving Method for Nonlinear Wave Equations on {R}iemannian Manifolds}, journal = {SIAM J. Sci. Comput.}, volume = {46}, number = {6}, pages = {A3779-A3802}, year = {2024}, doi = {10.1137/24M1654245}, } -
Simulating time-harmonic acoustic
wave effects induced by periodic holes/inclusions on surfaces.
Wen Hu, Zhuojia Fu, and Leevan Ling.
Applied Mathematical Modelling. 132:630-644, 2024.Abstract: This paper introduces a localized meshless method to analyze time-harmonic acoustic wave propagation on curved surfaces with periodic holes/inclusions. In particular, the generalized finite difference method is used as a localized meshless technique to discretize the surface gradient and Laplace-Beltrami operators defined extrinsically in the governing equations. An absorbing boundary condition is introduced to reduce reflections from boundaries and accurately simulate wave propagation on unclosed surfaces with periodic inclusions. Several benchmark examples demonstrate the efficiency and accuracy of the proposed method in simulating acoustic wave propagation on surfaces with diverse geometries, including complex shapes and periodic holes or inclusions.
@ARTICLE{Hu+Ling-Simutimeacouwave:24, author = {Hu, W. and Fu, Z. and Ling, L.}, title = {Simulating time-harmonic acoustic wave effects induced by periodic holes/inclusions on surfaces}, journal = {Applied Mathematical Modelling}, volume = {132}, pages = {630-644}, year = {2024}, doi = {10.1016/j.apm.2024.05.009}, } -
Realistic pattern formations on surfaces
by adding arbitrary roughness.
Siqing Li, Leevan Ling, Steve J. Ruuth, and Xuemeng Wang.
SIAM Journal on Applied Mathematics. 84(3):1163-1185, 2024.
Abstract: We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.
@ARTICLE{Li+LingETAL-Realpattformsurf:24, author = {Li, S. and Ling, L. and Ruuth, S. J. and Wang, X.}, title = {Realistic Pattern Formations on Surfaces by Adding Arbitrary Roughness}, journal = {SIAM Journal on Applied Mathematics}, volume = {84}, number = {3}, pages = {1163-1185}, year = {2024}, doi = {10.1137/22M1518001}, } -
Enhancing RBF-FD efficiency for
highly non-uniform nodes distributions via adaptivity.
Siqing Li, Leevan Ling, Xin Liu, Pankaj K. Mishra, Mrinal K. Sen, and Jing Zhang.
Numerical Mathematics: Theory, Methods and Applications. 17:331-350, 2024.Abstract: Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.
@ARTICLE{Li+LingETAL-EnhaRBF-effihigh:24, author = {Li, S. and Ling, L. and Liu, X. and Mishra, P. K. and Sen, M. K. and Zhang, J.}, title = {Enhancing {RBF-FD} Efficiency for Highly Non-Uniform Node Distributions via Adaptivity}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {2}, pages = {331--350}, doi = {10.4208/nmtma.OA-2023-0095}, } -
Exploring oversampling in RBF
least-squares collocation method of lines for surface diffusion.
Meng Chen, and Leevan Ling.
Numerical Algorithms. 97(3):1067-1087, 2024.Abstract: This paper investigates the numerical behavior of the radial basis functions least-squares collocation (RBF-LSC) method of lines (MoL) for solving surface diffusion problems, building upon the theoretical analysis presented in [Chen et al. SIAM J. Numer. Anal. 61(3), 1386-1404 (2023)]. Specifically, we examine the impact of the oversampling ratio, defined as the number of collocation points used over the number of RBF centers for quasi-uniform sets, on the stability of the eigenvalues, time-stepping sizes taken by Runge-Kutta methods, and overall accuracy of the method. By providing numerical evidence and insights, we demonstrate the importance of the oversampling ratio for achieving accurate and efficient solutions with the RBF-LSC-MoL method. Our results reveal that the oversampling ratio plays a critical role in determining the stability of the eigenvalues, and we provide guidelines for selecting an optimal oversampling ratio that balances accuracy and computational efficiency.
@ARTICLE{Chen+Ling-Exploverleascoll:24, author = {Chen, M. and Ling, L.}, title = {Exploring oversampling in {RBF} least-squares collocation method of lines for surface diffusion}, journal = {Numerical Algorithms}, year = {2024}, volume = {97}, number = {3}, pages = {1067-1087}, doi = {10.1007/s11075-023-01741-4}, } -
A kernel-based least-squares
collocation method for surface diffusion.
Meng Chen, Ka Chun Cheung, and Leevan Ling.
SIAM Journal on Numerical Analysis. 61(3): 1386-1404, 2023.Abstract: There are plenty of applications and analyses for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used asymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which are methods of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces.
@ARTICLE{Chen+CheungETAL-kernleascollmeth:23, author = {Chen, M. and Cheung, K. C. and Ling, L.}, title = {A Kernel-Based Least-Squares Collocation Method for Surface Diffusion}, journal = {SIAM J. Numer. Anal.}, volume = {61}, number = {3}, pages = {1386-1404}, year = {2023}, doi = {10.1137/21M1444369}, } -
A time-continuous embedding method for scalar hyperbolic conservation laws on manifolds
.
Yinghua Wang, Bao-Shan Wang, Leevan Ling, and Wai Sun Don.
Journal of Scientific Computing, 93:84, 2022.Abstract: A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in $\mathbb{R}^2$ and $\mathbb{R}^3$). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers' equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers' equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers' equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.
@ARTICLE{Wang+WangETAL-TimeEmbeMethScal:22, author = {Wang, Y. and Wang, B.-S. and Ling, L. and Don, W. S.}, title = {A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds}, journal = {J. Sci. Comput.}, year = {2022}, volume = {93}, number = {3}, pages = {84}, doi = {10.1007/s10915-022-02023-2}, } -
Solving interpolation problems
on surfaces stochastically and greedily.
Meng Chen, Leevan Ling, and Y. Su.
Dolomites Research Notes on Approximation, 15(3): 26-36, 2022.Abstract: Choosing suitable shape parameters in the kernel-based interpolation problems is an open question, whose solutions can guarantee accuracy and numerical stability. In this paper, we study various ways to select Kernel's shape parameters for interpolation problems on surfaces. In particular, we use exact and stochastically approximated cross validation approaches to select the shape parameters. When we solve the resultant matrix systems, we also deploy a greedy trial subspace selection algorithm to improve robustness. Numerical experiments are inserted along our discussion to demonstrate the feasibility and robustness of our proposed methods.
@ARTICLE{Chen+SuETAL-Solvinteprobsurf:22, AUTHOR = {Chen, M. and Ling, L. and Su, Y.}, TITLE = {Solving interpolation problems on surfaces stochastically and greedily}, YEAR = {2022}, JOURNAL = {Dolomites Research Notes on Approximation}, volume = {15}, number = {3}, pages = {26-36}, doi = {10.14658/pupj-drna-2022-3-4}, } -
Meshfree semi-Lagrangian methods
for solving surface advection PDEs.
Argyrios Petras, Leevan Ling, and Steve J. Ruuth.
Journal of Scientific Computing, 93:11, 2022.Abstract: We analyze a class of meshfree semi-Lagrangian methods for solving advection problems on smooth, closed surfaces with solenoidal velocity field. In particular, we prove the existence of an embedding equation whose corresponding semi-Lagrangian methods yield the ones in the literature for solving problems on surfaces. Our analysis allows us to apply standard bulk domain convergence theories to the surface counterparts. In addition, we provide detailed descriptions for implementing the proposed methods to run on point clouds. After verifying the convergence rates against the theory, we show that the proposed method is a robust building block for more complicated problems, such as advection problems with non-solenoidal velocity field, inviscid Burgers' equations and systems of reaction advection diffusion equations for pattern formation.
@ARTICLE{Petras+LingETAL-MeshSemiMethSolv:22, author = {Petras, A. and Ling, L. and Ruuth, S. J.}, title = {Meshfree Semi-{L}agrangian Methods for Solving Surface Advection {PDE}s}, journal = {J. Sci. Comput.}, year = {2022}, volume = {93}, pages = {11}, doi = {10.1007/s10915-022-01966-w}, } -
A kernel-based meshless conservative
Galerkin method for solving Hamiltonian wave equations.
Zhengjie Sun, and Leevan Ling.
SIAM Journal on Scientific Computing, 44(4): A2789--A2807, 2022.Abstract: We propose a meshless conservative Galerkin method for solving Hamiltonian wave equations. We first discretize the equation in space using radial basis functions in a Galerkin-type formulation. Differ from the traditional RBF Galerkin method that directly uses nonlinear functions in its weak form, our method employs appropriate projection operators in the construction of the Galerkin equation, which will be shown to conserve global energies. Moreover, we provide a complete error analysis to the proposed discretization. We further derive the fully discretized solution by a second order average vector field scheme. We prove that the fully discretized solution preserved the discretized energy exactly. Finally, we provide some numerical examples to demonstrate the accuracy and the energy conservation.
@ARTICLE{Sun+Ling-KernMeshConsGale:22, author = {Sun, Z. and Ling, L.}, title = {A Kernel-Based Meshless Conservative {G}alerkin Method for Solving {H}amiltonian Wave Equations}, journal = {SIAM J. Sci. Comput.}, volume = {44}, number = {4}, pages = {A2789-A2807}, year = {2022}, doi = {10.1137/21M1436981}, } -
A stochastic extended Rippa's
algorithm for LpOCV.
Leevan Ling, and Francesco Marchetti.
Applied Mathematics Letters. 129:107955, 2022.Abstract: In kernel-based approximation, the tuning of the so-called shape parameter is a fundamental step for achieving an accurate reconstruction. Recently, the popular Rippa's algorithm Rippa (1999) has been extended to a more general cross validation setting. In this work, we propose a modification of such extension with the aim of further reducing the computational costs. The resulting Stochastic Extended Rippa's Algorithm (SERA) is first detailed and then tested by means of various numerical experiments, which show its efficacy and effectiveness in different approximation settings.
@ARTICLE{Marchetti+Leevan-stocexteRippalgo:22, author = {Marchetti, F. and Ling, L.}, title = {A stochastic extended {R}ippa's algorithm for {LpOCV}}, journal = {Appl. Math. Lett.}, volume = {129}, pages = {107955}, year = {2022}, doi = {10.1016/j.aml.2022.107955}, } -
A localized extrinsic collocation
method for Turing pattern formations on surfaces.
Zhuochao Tang, Zhuojia Fu, Meng Chen, and Leevan Ling.
Applied Mathematics Letters. 122:107534, 2021Abstract: In this paper, we give our first attempt to implement a localized collocation method, namely Generalized Finite Difference Method (GFDM), for the Turing patterns formation problems on smooth, closed, connected surfaces of codimension one embedded in $\mathbb{R}^3$. Based on projections from surface differential operators to Euclidean differential operators, the surface PDEs in extrinsic form are given explicitly and could be solved directly by GFDM only using a set of collocation points distributed on surfaces. A sparse system formed from localization scheme makes it efficient for solving long time evolution Turing patterns formation problems. Numerical demonstrations including convergence test, Turing spot and stripe problems are provided to illustrate its potentiality.
@ARTICLE{Tang+FuETAL-locaextrcollmeth:21, author = {Tang, Z. and Fu, Z. and Chen, M. and Ling, L.}, title = {A localized extrinsic collocation method for {T}uring pattern formations on surfaces}, journal = {Appl. Math. Lett.}, volume = {122}, pages = {107534}, year = {2021}, doi = {10.1016/j.aml.2021.107534}, } -
On variable and random shape
Gaussian interpolations.
Sung Nok Chiu, Leevan Ling, and Michael McCourt.
Applied Mathematics and Computation. 377: 125159, 2020.Abstract: This work focuses on the invertibility of non-constant shape Gaussian asymmetric interpolation matrix, which includes the cases of both variable and random shape parameters. We prove a sufficient condition for that these interpolation matrices are invertible almost surely for the choice of shape parameters. The proof is then extended to the case of anisotropic Gaussian kernels, which is subjected to independent componentwise scalings and rotations. As a corollary of our proof, we propose a parameter free random shape parameters strategy to completely eliminate the need of users' inputs. By studying numerical accuracy in variable precision computations, we demonstrate that the asymmetric interpolation method is not a method with faster theoretical convergence. We show empirically in double precision, however, that these spatially varying strategies have the ability to outperform constant shape parameters in double precision computations. Various random distributions were numerically examined.
@ARTICLE{Chiu+LingETAL-varirandshapGaus:20, author = {Chiu, S. N. and Ling, L. and McCourt, M.}, title = {On variable and random shape {G}aussian interpolations}, journal = {Appl. Math. Comput.}, volume = {377}, pages = {125159}, year = {2020}, } -
Complex pattern formations by
spatial varying parameters.
Siqing Li, and Leevan Ling.
Advances in Applied Mathematics and Mechanics. 12(6): 1327-1352, 2020
Abstract: Pattern formations by Gierer-Meinhardt (GM) activator-inhibitor model are considered in this paper. By linear analysis, critical value of bifurcation parameter can be evaluated to ensure Turing instability. Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization. We numerically show the convergence of our algorithm. Pattern transitions in irregular domains are shown. We also provide various parameter settings on some irregular domains for different patterns appeared in nature. To further simulate patterns in reality, we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.
@ARTICLE{Li+Ling-CompPattFormSpat:20, author = { Li, S. and Ling, L.}, title = {Complex Pattern Formations by Spatial Varying Parameters}, journal = {Adv. Appl. Math. Mech.}, year = {2020}, volume = {12}, number = {6}, pages = {1327--1352}, issn = {2075-1354}, doi = {10.4208/aamm.OA-2018-0266}, } -
Extrinsic meshless collocation methods
for PDEs on manifolds.
Meng Chen, and Leevan Ling.
SIAM Journal on Numerical Analysis. 58(2):988-1007. 2020Abstract: We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, i.e., Kansa methods. Our main theoretical contribution is the robust convergence analysis under some standard smoothness assumptions for high-order convergence. Numerical demonstrations are provided to verify the proven convergence rates, and we simulate reaction-diffusion equations for generating Turing patterns on manifolds.
@ARTICLE{Chen+Ling-Extrmeshcollmeth:20, author = {Chen, M. and Ling, L.}, title = {Extrinsic meshless collocation methods for {PDE}s on manifolds}, year = {2020}, journal = {SIAM J. Numer. Anal.}, volume = {58}, number = {2}, pages = {988-1007}, doi = {10.1137/17M1158641}, } -
Kernel-based collocation methods
for heat transport on evolving surfaces.
Meng Chen, and Leevan Ling.
Journal of Computational Physics. 405: 109166, 2020.Abstract: We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on extrinsic kernel-based meshless collocation methods. The algorithms can be classified into two categories: one collocates PDEs extrinsically and analytically, and the other approximates surface differential operators by meshless pseudospectral approaches. The former is specifically designed to handle PDEs on evolving surfaces defined by parametric equations, and the latter works on surface evolutions based on point clouds. After some convergence studies and comparisons, we demonstrate that the proposed method can solve challenging PDEs posed on surfaces with high curvatures with discontinuous initial conditions with correct physics.
@ARTICLE{Chen+Ling-Kerncollmethheat:20, author = {Chen, M. and Ling, L.}, title = {Kernel-based collocation methods for heat transport on evolving surfaces.}, journal = {J. Comput. Phys.}, volume = {405}, pages = {109166}, year = {2020}, doi = {10.1016/j.jcp.2019.109166}, } -
Solving Partial Differential
Equations on Surfaces with Fundamental Solutions.
Meng Chen, Ka Chun Cheung, and Leevan Ling.
In: Alves C., Karageorghis A., Leitão V., Valtchev S. (eds) Advances in Trefftz Methods and Their Applications. SEMA SIMAI Springer Series, Springer, Cham. 23:1-11, 2020.Abstract: The aim of this paper is to present partial differential equations (PDEs) on surface to the community of methods of fundamental solutions (MFS). First, we present an embedding formulation to embed surface PDEs into a domain so that MFS can be applied after the PDEs is homogenized with a particular solution. Next, we discuss how the domain-MFS method can be used to directly collocate surface PDEs. Some numerical demonstrations were included to study the effect of basis functions and source point locations.
@INPROCEEDINGS{Chen+CheungETAL-SolvPartDiffEqua:20, author = {Chen, M. and Cheung, K. C. and Ling, L.}, editor = {Alves, Carlos and Karageorghis, Andreas and Leit{\~a}o, Vitor and Valtchev, Svilen}, title = {Solving Partial Differential Equations on Surfaces with Fundamental Solutions}, booktitle = {Advances in Trefftz Methods and Their Applications}, year = {2020}, publisher = {Springer International Publishing}, address = {Cham}, pages = {1--11}, doi = {10.1007/978-3-030-52804-1_1}, } -
Meshfree seismic modeling using
radial basis finite-difference with adaptive stencil size.
Xin Liu, Pankaj K. Mishra, Mrinal K. Sen, and Leevan Ling.
SEG Technical Program Expanded Abstracts. 2673-2677, 2020.Abstract: In a meshfree method, the solution of a partial differential equation (PDE) can be approximated over scattered and possibly non-uniform nodes depending on the spatial distribution of the model parameter in the domain. Conventionally, a fixed stencil size is applied to the whole computational domain with non-uniform node-distribution, making the accuracy of the method higher in high-density node region than the accuracy in low-density node regions in the domain. In this work, we implement meshfree seismic wave modeling with variable stencil-size Radial Basis Functions-generated Finite-difference method (RBF-FD) to reduce the redundancy and enhance computational efficiency. The proposed method employs smaller stencil sizes at shorter fill distance regions compared to stencil sizes at the longer fill distance regions. The variable stencil sizes are selected such that the global accuracy is consistent for the whole model.
@INBOOK{Liu+MishraETAL-Meshseismodeusin:20, author = {Liu, X. and Mishra, P. K. and Sen, M. K. and Ling, L.}, title = {Meshfree seismic modeling using radial basis finite-difference with adaptive stencil size}, booktitle = {SEG Technical Program Expanded Abstracts 2020}, pages = {2673-2677}, year = {2020}, publisher = {Society of Exploration Geophysicists}, doi = {10.1190/segam2020-3428393.1}, } -
Meshless collocation methods for solving PDEs on surfaces.
Meng Chen, Ka Chun Cheung, and Leevan Ling.
In A.H.-D. Cheng, A. Tadeu (Eds.), WIT Transactions on Engineering Sciences, 126, 159-170, 2019.Abstract: We present three recently proposed kernel-based collocation methods in unified notations as an easy reference for practitioners who need to solve PDEs on surfaces $S\subset\mathbb{R}^d$. These PDEs closely resemble their Euclidean counterparts, except that the problem domains change from bulk regions with a flat geometry of some surfaces, on which curvatures play an important role in the physical processes. First, we present a formulation to solve surface PDEs in a narrow band domain containing the surface. This class of numerical methods is known as the embedding types. Next, we present another formulation that works solely on the surface, which is commonly referred to as the intrinsic approach. Convergent estimates and numerical examples for both formulations will be given. For the latter, we solve both the linear and nonlinear time-dependent parabolic equations on static and moving surfaces.
@ARTICLE{CHEN+CHEUNGETAL-Meshcollmethsolv:19, title = {Meshless collocation methods for solving {PDE}s on surfaces}, author = {Chen, M. and Cheung, K. C. and Ling, L.}, journal = {WIT Trans. Eng. Sci.}, volume = {126}, pages = {159--170}, year = {2019}, publisher = {WIT Press}, } -
Kernel-based meshless collocation
methods for solving coupled bulk-surface PDEs.
Meng Chen, and Leevan Ling.
Journal of Scientific Computing. 81(1): 375-391, 2019.Abstract: A meshless kernel-based method is developed to solve coupled second-order elliptic PDEs in bulk domains and surfaces, subject to Robin boundary conditions. It combines a least-squares kernel collocation method with a surface-type intrinsic approach. Therefore, we can use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, for the search of least-squares solutions in bulks and on surfaces respectively. We first give error estimates for domain-type Robin-boundary problems. Based on this and existing results for surface PDEs, we discuss the theoretical requirements for the employed Sobolev kernels. Then, we select the orders of smoothness for the kernels in bulks and on surfaces. Lastly, several numerical experiments are demonstrated to test the robustness of the coupled method for accuracy and convergence rates under different settings.
@ARTICLE{Chen-KernMeshCollMeth:19, author = {Chen, M. and Ling, L.}, title = {Kernel-Based Meshless Collocation Methods for Solving Coupled Bulk--Surface Partial Differential Equations}, journal = {J. Sci. Comput.}, year = {2019}, volume = {81}, number = {1}, pages = {375--391}, doi = {10.1007/s10915-019-01020-2}, } -
Collocation methods for Cauchy
problems of elliptic operators via conditional stabilities.
Siqing Li, and Leevan Ling.
Communications in Computational Physics. 26(3): 785-808, 2019.Abstract: Ill-posed Cauchy problems for elliptic partial differential equations appear in many engineering fields. In this paper, we focus on stable reconstruction methods for this kind of inverse problems. Using kernels that reproduce Hilbert spaces $H^m(\Omega)$, numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints (LSQI). A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm using generalized singular value decomposition of matrices and Lagrange multipliers is proposed to solve the LSQI problem. Numerical experiments of two-dimensional cases verify our proved convergence results. By comparing with solutions of MFS and FEM under the discrete Tikhonov regularization by RKHS under same Cauchy data, we demonstrate that our method can reconstruct stable and high accuracy solutions for noisy Cauchy data.
@ARTICLE{Li+Ling-CollMethCaucProb:19, author = {Li, S. and Ling, L.}, title = {Collocation Methods for {C}auchy Problems of Elliptic Operators via Conditional Stabilities}, journal = {Commun. Comput. Phys.}, volume = {26}, number = {3}, pages = {785-808}, year = {2019}, doi = {10.4208/cicp.OA-2018-0182}, } -
Weighted least-squares
collocation methods for elliptic PDEs with mixed boundary conditions.
Siqing Li, and Leevan Ling.
Engineering Analysis with Boundary Elements. 105: 146-154, 2019.Abstract: In this paper, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on other three factors: the proportion of measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved convergent in $H^2(\Omega)$. Numerical experiments for two dimensional examples show that we can obtain convergent solutions for kernel smoothness $m\in\{3, 4, 5\}$ in the irregular domains, circle domain, and rectangle thin domain. We also apply our formulations to three dimensional cases and get desired convergent results for $m\in\{4, 5, 6, 7\}$ in cubic, sphere and torus domain under different boundary conditions.
@ARTICLE{Li+Ling-Weigleascollmeth:19, author = {Li, S. and Ling, L.}, title = {Weighted least-squares collocation methods for elliptic {PDE}s with mixed boundary conditions}, journal = {Eng. Anal. Bound. Elem.}, volume = {105}, pages = {146-154}, year = {2019}, doi = {10.1016/j.enganabound.2019.04.012}, } -
A stabilized radial basis-finite
difference (RBF-FD) method with hybrid kernels.
Pankaj K. Mishra, Gregory E. Fasshauer, Mrinal K. Sen and Leevan Ling.
Computers and Mathematics with Applications. 77(9): 2354-2368, 2019.Abstract: Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.
@ARTICLE{Mishra+FasshauerETAL-stabradibasidiff:19, author = {Mishra, P. K. and Fasshauer, G. E. and Sen, M. K. and Ling, L. }, title = {A stabilized radial basis-finite difference ({RBF-FD}) method with hybrid kernels}, journal = {Comput. Math. Appl.}, volume = {77}, number = {9}, pages = {2354-2368}, year = {2019}, doi = {10.1016/j.camwa.2018.12.027}, } -
Discrete least-squares radial basis
functions approximations.
Siqing Li, Leevan Ling, and Ka Chun Cheung.
Applied Mathematics and Computation. 355: 542-552, 2019.Abstract: We consider discrete least-squares methods using radial basis functions. A general $\ell^2$-Tikhonov regularization with $W_2^m$ penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases in which the function it is approximating is within and is outside of the native space of the kernel. Our proven theories concern the denseness condition of collocation points and selection of regularization parameters. In particular, the unregularized least-squares method is shown to have $W_2^\mu(\Omega)$ convergence for $\mu>d/2$ on a smooth domain $\Omega\subset\mathbb{R}^d$. For any properly regularized least-squares method, the same convergence estimates hold for a large range of $\mu\geq 0$. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In terms of applications, we also apply the proposed method to solve a heat equation whose initial condition has huge oscillation in the domain.
@ARTICLE{Li+CheungETAL-Discleasradibasi:19, author = {Li, S.and Ling, L. and Cheung, K. C. }, title = {Discrete least-squares radial basis functions approximations}, journal = {Appl. Math. Comput.}, volume = {355}, pages = {542-552}, year = {2019}, doi = {10.1016/j.amc.2019.03.007}, } -
A least-squares implicit RBF-FD closest
point method and applications to PDEs on moving surfaces.
Argyrios Petras, Leevan Ling, Cecile Piret, and Steve J. Ruuth.
Journal of Computational Physics. 381: 146-161, 2019.Abstract: The closest point method (Ruuth and Merriman (2008) [32]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao (2009) [37]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.
@ARTICLE{Petras+LingETAL-leasimplRBF-clos:19, author = {Petras, A. and Ling, L. and Piret, C. and Ruuth, S. J.}, title = {A least-squares implicit {RBF-FD} closest point method and applications to {PDE}s on moving surfaces}, journal = {J. Comput. Phys.}, volume = {381}, pages = {146-161}, year = {2019}, doi = {10.1016/j.jcp.2018.12.031}, } -
An RBF-FD closest point method for
solving PDEs on surfaces.
Argyrios Petras, Leevan Ling, and Steve J. Ruuth.
Journal of Computational Physics. 370: 43-57, 2018.Abstract: Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008)) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012)), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009)). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
@ARTICLE{Petras+LingETAL-RBF-clospoinmeth:18, author = {Petras, A. and Ling, L. and Ruuth, S. J.}, title = {An {RBF-FD} closest point method for solving {PDE}s on surfaces}, journal = {J. Comput. Phys.}, volume = {370}, pages = {43-57}, year = {2018}, doi = {10.1016/j.jcp.2018.05.022}, } -
Doubly stochastic radial basis
function methods.
Fenglian Yang, Liang Yan, and Leevan Ling.
Journal of Computational Physics. 363: 87-97, 2018.Abstract: We propose a doubly stochastic radial basis function (DSRBF) method for function recoveries. Instead of a constant, we treat the RBF shape parameters as stochastic variables whose distribution were determined by a stochastic leave-one-out cross validation (LOOCV) estimation. A careful operation count is provided in order to determine the ranges of all the parameters in our methods. The overhead cost for setting up the proposed DSRBF method is $O(n^2)$ for function recovery problems with n basis. Numerical experiments confirm that the proposed method not only outperforms constant shape parameter formulation (in terms of accuracy with comparable computational cost) but also the optimal LOOCV formulation (in terms of both accuracy and computational cost).
@ARTICLE{Yang+YanETAL-Doubstocradibasi:18, author = {Yang, F. L. and Yan, L. and Ling, L.}, title = {Doubly stochastic radial basis function methods}, journal = {J. Comput. Phys.}, volume = {363}, pages = {87-97}, year = {2018}, doi = {10.1016/j.jcp.2018.02.042}, } -
Evaluation finite moment
log-stable option pricing by a spectral method.
Xu Guo, and Leevan Ling.
Numerical Mathematics: Theory, Methods and Applications. 11(3): 437-452, 2018.Abstract: The classical Black-Scholes pricing model is based on standard geometric Brownian motion, and the log-returns of this model are independent and Gaussian. However, most of the recent researches on the statistical properties of the log-returns make this hypothesis not always consistent. One of the ongoing issues of mathematical finance today is to design an efficient numerical algorithm for the pricing model, which might be modified from the standard Black-Scholes diffusion equation and would have favorable empirical results. Of those financial models that have been already proposed, the most interesting include the Finite Moment Log-Stable (FMLS) process model and its fractional partial integral-differential equation. In this paper, we consider using Gauss-Jacobi spectral method on a two-dimensional computation domain in order to discretize the FMLS fractional partial integral-differential equation, and further illustrate the flexibility and accuracy of the method by comparing the first order finite difference scheme for the pricing examples of European and American-styled options. Our results suggest that the global character of the Gauss-Jacobi method makes them well-suited to fractional partial integral-differential equations and can naturally take the global behavior of the solution into account and thus do not lead to an extra computational cost when moving from a second-order to a fractional-order diffusion model.
@ARTICLE{Xu+Ling-EvalFiniMomeLog-:18, author = {Guo, X. and Ling, L.}, title = {Evaluation Finite Moment Log-Stable Option Pricing by a Spectral Method}, journal = {Numer. Math. Theor. Meth. Appl.}, volume = {11}, number = {3}, pages = {437-452}, year = {2018}, doi = {10.4208/nmtma.2017-OA-0131}, } -
On meshfree numerical
differentiation.
Leevan Ling, and Qi Ye.
Analysis and Applications. 16(5): 717-739, 2018.Abstract: We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimators from normal random variables, which are defined by kernel-based probability measures induced from symmetric positive definite kernels, to reconstruct the unknown partial derivatives from scattered noisy data. Our developed theories give rise to Tikhonov regularization methods with a priori parameter, but the shape parameters of the kernels remain tunable. For that, we propose an error measure that is computable without the exact values of the derivative. This allows users to obtain a quasi-optimal kernel-based estimator by comparing the approximation quality of kernel-based estimators. Numerical examples in two dimensions and three dimensions are included to demonstrate the convergence behavior and effectiveness of the proposed numerical differentiation scheme.
@ARTICLE{Ling+Ye-meshnumediff:18, author = {Ling, L. and Ye, Q.}, title = {On meshfree numerical differentiation}, journal = {Anal. Appl.}, volume = {16}, number = {5}, pages = {717-739}, year = {2018}, doi = {10.1142/S021953051850001X}, } -
Fully adaptive kernel-based
methods.
Leevan Ling, and Sung Nok Chiu.
International Journal for Numerical Methods in Engineering. 114(4): 454-467, 2018.Abstract: By exploiting the meshless property of kernel-based collocation methods, we propose a fully automatic numerical recipe for solving interpolation/regression and boundary value problems adaptively. The proposed algorithm is built upon a least squares collocation formulation on some quasi-random point sets with low discrepancy. A novel strategy is proposed to ensure that the fill distances of data points in the domain and on the boundary are in the same order of magnitude. To circumvent the potential problem of ill-conditioning due to extremely small separation distance in the point sets, we add an extra dimension to the data points for generating shape parameters such that nearby kernels are of distinctive shape. This effectively eliminates the needs of shape parameter identification. Resulting linear systems were then solved by a greedy trial space algorithm to improve the robustness of the algorithm. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed methods.
@ARTICLE{Ling+Chiu-Fulladapkernmeth:18, author = {Ling, L. and Chiu, S. N.}, title = {Fully adaptive kernel-based methods}, journal = {Int. J. Numer. Methods Eng.}, volume = {114}, number = {4}, pages = {454-467}, year = {2018}, doi = {10.1002/nme.5750}, } -
A kernel-based embedding method and
convergence analysis for surfaces PDEs.
Ka Chun Cheung, and Leevan Ling.
SIAM Journal on Scientific Computing. 40(1): A266-A287, 2018.Abstract: We analyze a least-squares strong-form kernel collocation formulation for solving second-order elliptic PDEs on smooth, connected, and compact surfaces with bounded geometry. The methods do not require any partial derivatives of surface normal vectors or metric. Based on some standard smoothness assumptions for high-order convergence, we provide the sufficient denseness conditions on the collocation points to ensure the methods are convergent. In addition to some convergence verifications, we also simulate some reaction-diffusion equations to exhibit the pattern formations.
@ARTICLE{Cheung+Ling-Kernembemethconv:18, AUTHOR = {Cheung, K. C. and Ling, L.}, TITLE = {A kernel-based embedding method and convergence analysis for surfaces {PDE}s}, journal = {SIAM J. Sci. Comput.}, volume = {40}, number = {1}, pages = {A266-A287}, year = {2018}, doi = {10.1137/16M1080410}, } -
H2-convergence of
least-squares kernel collocation methods.
Ka Chun Cheung, Leevan Ling, and Robert Schaback.
SIAM Journal on Numerical Analysis. 56(1): 614-633, 2018.
Abstract: The strong-form asymmetric kernel-based collocation method, commonly referred to as the Kansa method, is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support. The simple least-squares (LS) formulation, on the other hand, makes the study of its solvability and convergence rather nontrivial. In this paper, we focus on general second order linear elliptic differential equations in $\Omega \subset \mathbb{R}^d$ under Dirichlet boundary conditions. With kernels that reproduce $H^m(\Omega)$ and some smoothness assumptions on the solution, we provide conditions for a constrained LS method and a class of weighted LS algorithms to be convergent. Theoretically, for $\max\left(2, \lceil (d+1)/2 \rceil\right) \le \nu \le m$, we identify some $H^\nu(\Omega)$ convergent LS formulations that have an optimal error behavior like $h^{m-\nu}$. For $d \le 3$, the proposed methods are optimal in $H^2(\Omega)$. We demonstrate the effects of various collocation settings on the respective convergence rates.
@ARTICLE{Cheung+LingETAL-leaskerncollmeth:18, author = {Cheung, K. C. and Ling, L. and Schaback, R.}, title = {${H}^2$--Convergence of least-squares kernel collocation methods}, journal = {SIAM J. Numer. Anal.}, volume = {56}, number = {1}, pages = {614-633}, year = {2018}, doi = {10.1137/16M1072863}, } -
Convergence studies for an
adaptive meshless least-squares collocation method.
Ka Chun Cheung, and Leevan Ling.
International Journal of Computational Methods and Experimental Measurements. 5(3): 377-386, 2017.Abstract: In this paper, we apply the recently proposed fast block-greedy algorithm to a convergent kernel-based collocation method. In particular, we discretize three-dimensional second-order elliptic differential equations by the meshless asymmetric collocation method with over-sampling. Approximated solutions are obtained by solving the resulting weighted least squares problem. Such formulation has been proven to have optimal convergence in $H^2$. Our aim is to investigate the convergence behaviour of some three dimensional test problems. We also study the low-rank solution by restricting the approximation in some smaller trial subspaces. A block-greedy algorithm, which costs at most $\mathcal{O}(NK^2)$ to select $K$ columns (or trial centers) out of an $M \times N$ overdetermined matrix, is employed for such an adaptivity. Numerical simulations are provided to justify these reductions.
@ARTICLE{Cheung+Ling-Convstudadapmesh:17, AUTHOR = {Cheung, K. C. and Ling, L.}, TITLE = {Convergence studies for an adaptive meshless least-squares collocation method}, YEAR = {2017}, JOURNAL = {Int. J. Comput. Methods Experimental Measurements.}, volume = {5}, number = {3}, pages = {377-386}, doi = {10.2495/CMEM-V5-N3-377-386}, } -
Meshless collocation
methods with graph Laplacian for PDEs on folded surface.
Ka Chun Cheung, and Leevan Ling.
Neural, Parallel, and Scientific Computations. 25: 79-90, 2017.Abstract: This work aims to provide a numerical treatment to deal with PDEs on surfaces with singularity along smooth curves. Previously, we proposed a low-order localized meshless method for discretizing the surface Laplacian operator on surfaces with folded. It is observed that the previously proposed strategy of approximating surface Laplacian at fold does not work with a recently developed high-order embedding meshless method for smooth surface. In this paper, we propose using the graph Laplacian to handle the surface singularity and study how it can be used to solve PDEs on folded surfaces.
@ARTICLE{Cheung+Ling-Meshcollmethwith:17, AUTHOR = {Cheung, K. C. and Ling, L.}, TITLE = {Meshless collocation methods with graph {L}aplacian for {PDE}s on folded surface}, YEAR = {2017}, JOURNAL = {Neural, Parallel, Sci. Comput.}, volume = {25}, pages = {79-90}, } -
Numerical
investigation on the effect of tumor on the thermal behavior inside the skin tissue.
Zhuojia Fu, Qiang Xi, Leevan Ling, and Chang-Yong Cao.
International Journal of Heat and Mass Transfer. 108, Part A, 1154-1163, 2017.Abstract: This paper investigates the thermal behavior inside skin tissue with the presence of a tumor by solving the Pennes bioheat equation. We apply the method of approximate particular solutions (MAPS) to simulate tumor in 3D. The MAPS is a kind of domain-type meshless collocation methods, which has the merits being free of mesh generation and numerical integration. To ensure the interface conditions were exactly satisfied, an affine space decomposition technique is adopted. After verifying the convergence of the proposed method, we provide various simulations to the thermal effects inside the skin tissue due to an arbitrary-shaped tumor, including the location, geometry and size.
@ARTICLE{Fu+XiETAL-Numeinveeffetumo:17, author = {Fu, Z. and Xi, Q. and Ling, L. and Cao, C.-Y.}, title = {Numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue }, journal = {Int. J. Heat Mass Transf.}, volume = {108, Part A}, pages = {1154 - 1163}, year = {2017}, doi = {10.1016/j.ijheatmasstransfer.2016.11.109}, } -
A fractional model for time-variant
non-Newtonian flow.
Xu Yang, Wen Chen, Rui Xiao, and Leevan Ling.
Thermal Science. 21(1A): 61-68, 2017.
Abstract: This work applies a fractional flow model to describe a time-variant behavior of non-Newtonian substances. Specifically, we model the physical mechanism underlying the thixotropic and anti-thixotropic phenomena of non-Newtonian flow. This study investigates the behaviors of cellulose suspensions and SMS pastes under constant shear rate. The results imply that the presented model with only two parameters is adequate to fit experimental data. Moreover, the parameter of fractional order is an appropriate index to characterize the state of given substances. Its value indicates the extent of thixotropy and anti-thixotropy with positive and negative order respectively.
@ARTICLE{Yang+ChenETAL-fracmodetimenon-:17, AUTHOR = {Yang, X. and Chen, W. and Xiao, R. and Ling, L.}, TITLE = {A fractional model for time-variant non-Newtonian flow}, year = {2017}, journal = {Thermal Sci.}, volume = {21}, number = {1A}, pages = {61-68}, doi = {10.2298/TSCI160426245Y}, } -
A fast block-greedy algorithm for
quasi-optimal meshless trial subspace selection.
Leevan Ling.
SIAM Journal on Scientific Computing. 38(2): A1224-A1250, 2016.Abstract: Meshless collocation methods are often seen as a flexible alternative to overcome difficulties that may occur with other methods. As various meshless collocation methods gain popularity, finding appropriate settings becomes an important open question. Previously, we proposed a series of sequential-greedy algorithms for selecting quasi-optimal meshless trial subspaces that guarantee stable solutions from meshless methods, all of which were designed to solve a more general problem: ``Let $A$ be an $M \times N$ matrix with full rank $M$; choose a large $M \times K$ submatrix formed by $K \le M$ columns of $A$ such that it is numerically of full rank.'' In this paper, we propose a block-greedy algorithm based on a primal/dual residual criterion. Similar to all algorithms in the series, the block-greedy algorithm can be implemented in a matrix-free fashion to reduce the storage requirement. Most significantly, the proposed algorithm reduces the computational cost from the previous $\mathcal{O}(K^4 + NK^2)$ to at most $\mathcal{O}(NK^2)$. Numerical examples are given to demonstrate how this efficient and ready-to-use approach can benefit the stability and applicability of meshless collocation methods.
@ARTICLE{Ling-fastblocalgoquas:16, AUTHOR = {Ling, L.}, TITLE = {A fast block-greedy algorithm for quasi-optimal meshless trial subspace selection}, year = {2016}, journal = {SIAM J. Sci. Comput.}, volume = {38}, number = {2}, pages = {A1224--A1250}, doi = {10.1137/15M1037627}, } -
Regularization for 2-D
fractional sideways heat equations.
Caixian Wang, Leevan Ling, Xiangtuan Xiong, and Ming Li.
Numerical Heat Transfer Fundamentals. 68(5):418-433, 2015
Abstract: In this article, an inverse problem of Caputo-time-fractional sideways heat equations is considered. The aim is to find the inaccessible boundary data of some heterogeneous materials through interior measurements. Instead of the standard Tikhonov approach, we propose a series of fast filters for reconstructing the missing boundary data. Theoretical error bounds are provided for both boundary and (near-boundary) interior reconstructions. Several numerical examples are included to verify the proven convergence results.
@ARTICLE{Wang+LingETAL-REGUFRACSIDEHEAT:15, author = {Wang, C. and Ling, L. and Xiong, X. and Li, M.}, title = {Regularization for 2-{D} fractional sideways heat equations}, journal = {Numer. Heat Tr. B-Fund.}, year = {2015}, volume = {68}, number = {5}, pages = {418-433}, doi = {10.1080/10407790.2015.1036629}, } -
A localized meshless method for
diffusion on folded surfaces.
Ka Chun Cheung, Leevan Ling, and Steve J. Ruuth.
Journal of Computational Physics. 297: 194-206, 2015Abstract: Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we propose a radial basis function (RBF) discretization of the closest point method. The corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces. Our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution. A stable solver is used to avoid the ill-conditioning that arises when the radial basis functions (RBFs) become flat.
@ARTICLE{Cheung+LingETAL-locameshmethdiff:15, author = {Cheung, K. C. and Ling, L. and Ruuth, S. J.}, title = {A localized meshless method for diffusion on folded surfaces}, journal = {J. Comput. Phys.}, volume = {297}, pages = {194-206}, year = {2015}, doi = {10.1016/j.jcp.2015.05.021}, } -
Method of approximate
particular solutions for constant- and variable-order fractional diffusion models.
Zhuojia Fu, Wen Chen, and Leevan Ling.
Engineering Analysis with Boundary Elements. 57: 37-46, 2015.
Abstract: The method of approximate particular solutions (MAPS) is an alternative radial basis function (RBF) meshless method, which is defined in terms of a linear combination of the particular solutions of the inhomogeneous governing equations with traditional RBFs as the source term. In this paper, we apply the MAPS to both constant- and variable-order time fractional diffusion models. In the discretization formulation, a finite difference scheme and the MAPS are used respectively to discretize time fractional derivative and spatial derivative terms. Numerical investigation examples show the present meshless scheme has highly accuracy and computationally efficiency for various fractional diffusion models.
@ARTICLE{Fu+ChenETAL-Methapprpartsolu:15, author = {Fu, Z.-J. and Chen, W. and Ling, L.}, title = {Method of approximate particular solutions for constant- and variable-order fractional diffusion models}, journal = {Eng. Anal. Bound. Elem.}, year = {2015}, volume = {57}, pages = {37-46}, doi = {10.1016/j.enganabound.2014.09.003}, } -
Numerical Caputo differentiation
by radial basis functions.
Ming Li, Yujiao Wang, and Leevan Ling.
Journal of Scientific Computing. 62(1):300-315, 2015.Abstract: Previously, based on the method of (radial powers) radial basis functions, we proposed a procedure for approximating derivative values from one-dimensional scattered noisy data. In this work, we show that the same approach also allows us to approximate the values of (Caputo) fractional derivatives (for orders between 0 and 1). With either an a priori or a posteriori strategy of choosing the regularization parameter, our convergence analysis shows that the approximated fractional derivative values converge at the same rate as in the case of integer order 1.
@ARTICLE{Li+WangETAL-NumeCapuDiffRadi:15, author = {Li, M. and Wang, Y. and Ling, L.}, title = {Numerical {C}aputo Differentiation by Radial Basis Functions}, journal = {J. Sci. Comput.}, year = {2015}, volume = {62}, number = {1}, pages = {300-315}, doi = {10.1007/s10915-014-9857-6}, } -
Collocation and optimization initialization.
Edward J. Kansa, and Leevan Ling.
In A.H-D. Cheng, C. A. Brebbia (Eds.), WIT Transactions on Modelling and Simulation, 57, 55-61, 2014.Abstract: Integrated volumetric methods such as finite elements and their ``meshless'' variations are typically smoother than the strong form finite difference and radial basis function collocation methods. Numerical methods decrease their convergence rates with successively higher orders of differentiation along with improved conditioning. In contrast, increasing order of integration increases the convergence rate at the expense of poorer conditioning. In the study presented, a two-dimensional Poisson equation with exponential dependency is solved. The solution of the point collocation problem becomes the initial estimate for an integrated volumetric minimization process. Global, rather than local integration, is used since there is no need to construct any meshes for integration as done in the “meshless” finite element analogs. The root mean square (RMS) errors are compared. By pushing the shape parameter to very large values, using extended precision, the RMS errors show that spatial refinement benefits are relatively small compared to pushing shape parameters to increasing larger values. The improved Greedy Algorithm was used to optimize the set of data and evaluation centers for various shape parameters. Finally, extended arithmetic precision is used to push the range of the shape parameters.
@ARTICLE{Kansa+Ling-CollOptiInit:14, title = {Collocation And Optimization Initialization}, author = {Kansa, E. J. and Ling, L.}, journal = {WIT Trans. Modelling Simulation}, volume = {57}, pages = {55 - 61}, year = {2014}, publisher = {WIT Press}, } -
Solving moving-boundary
problems with the wavelet adaptive radial basis functions method.
Leopold Vrankar, Nicolas Ali Libre, Leevan Ling, Goran Turk, and Franc Runovc.
Computers and Fluids. 86(5):37-44, 2013.Abstract: Moving boundaries are associated with the time-dependent problems where the momentary position of boundaries needs to be determined as a function of time. The level set method has become an effective tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, computer animation and image processing. This work extends our earlier work on solving moving boundary problems with adaptive meshless methods. In particular, the objective of this paper is to investigate numerical performance the radial basis functions (RBFs) methods, with compactly supported basis and with global basis, coupled with a wavelet node refinement technique and a greedy trial space selection technique. Numerical simulations are provided to verify the effectiveness and robustness of RBFs methods with different adaptive techniques.
@ARTICLE{Vrankar+LibreETAL-Solvmoviprobwith:13, author = {Vrankar, L. and Libre, N. A. and Ling, L. and Turk, G. and Runovc, F.}, title = {Solving moving-boundary problems with the wavelet adaptive radial basis functions method}, journal = {Comput. Fluids}, year = {2013}, volume = {86}, number = {5}, pages = {37-44}, doi = {10.1016/j.compfluid.2013.06.029}, } -
Convergent overdetermined-RBF-MLPG
for solving second order elliptic PDEs.
Ahmad Shirzadi, and Leevan Ling.
Advances in Applied Mathematics and Mechanics. 5(1):78-89, 2013.
Abstract: This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin (MLPG) method with radial basis function (RBF) kernels generated trial spaces. Local weak-form testings are done with step-functions. It is proved that subject to sufficiently many appropriate testings, solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed. Moreover, an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation. Numerical results (in double precision) give good agreement with the provided theory.
@ARTICLE{Shirzadi+Ling-ConvOverSolvSeco:13, author = {Shirzadi, A. and Ling, L.}, title = {Convergent Overdetermined-{RBF}-{MLPG} for Solving Second Order Elliptic {PDE}s}, journal = {Adv. Appl. Math. Mech.}, volume = {5}, number = {1}, pages = {78-89}, issn = {2070-0733}, year = {2013}, doi = {10.4208/aamm.11-m11168}, } -
Numerical simulations for
space-time fractional diffusion equations.
Leevan Ling, and Masahiro Yamamoto.
International Journal of Computational Methods. 10(2):1341001, 2013.Abstract: We consider the solutions of a space-time fractional diffusion equation on the interval $[-1, 1]$. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann-Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space-time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.
@ARTICLE{Ling+Yamamoto-Numesimuspacfrac:12, author = {Ling, L. and Yamamoto, M.}, title = {Numerical Simulations For Space-Time Fractional Diffusion Equations}, journal = {Int. J. Comput. Methods}, volume = {10}, number = {2}, pages = {1341001}, issn = {0219-8762}, year = {2013}, doi = {10.1142/s0219876213410016}, } -
Combinations of the method of
fundamental solutions for general inverse source identification problems.
Fuzhang Wang, Wen Chen, and Leevan Ling.
Applied Mathematics and Computation. 219(3):1173-1182, 2012.Abstract: In this paper, a new general scheme, based on the method of fundamental solutions, is presented for inverse source identification problems. This is fulfilled by coupling a linear combination of fundamental solutions and radial basis functions associated with particular solutions. Under this scheme, we can determine harmonic and nonharmonic source terms from partially accessible boundary measurements. Numerical results for several general inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.
@ARTICLE{Wang+ChenETAL-Combmethfundsolu:12, author = {Wang, F. and Chen, W. and Ling, L.}, title = {Combinations of the method of fundamental solutions for general inverse source identification problems}, year = {2012}, journal = {Appl. Math. Comput.}, volume = {219}, number = {3}, pages = {1173-1182}, doi = {10.1016/j.amc.2012.07.027} } -
Meshless simulations of
the two-dimensional fractional-time convection-diffusion-reaction equations.
Ahmad Shirzadi, Leevan Ling, and S. Abbasbandy.
Engineering Analysis with Boundary Elements. 36(11):1522-1527, 2012.Abstract: The aim of this work is to propose a numerical approach based on the local weak formulations and finite difference scheme to solve the two-dimensional fractional-time convection-diffusion-reaction equations. The numerical studies on sensitivity analysis to parameter and convergence analysis show that our approach is stable. Moreover, numerical demonstrations are given to show that the weak-form approach is applicable to a wide range of problems; in particular, a forced-subdiffusion-convection equation previously solved by a strong-form approach with weak convection is considered. It is shown that our approach can obtain comparable simulations not only in weak convection but also in convection dominant cases. The simulations to a subdiffusion-convection-reaction equation are also presented.
@ARTICLE{Shirzadi+LingETAL-Meshsimutwo-frac:12, author = {Shirzadi, A. and Ling, L. and Abbasbandy, S.}, title = {Meshless simulations of the two-dimensional fractional-time convection-diffusion-reaction equations}, journal = {Eng. Anal. Bound. Elem.}, volume = {36}, number = {11}, pages = {1522-1527}, year = {2012}, doi = {10.1016/j.enganabound.2012.05.005} } -
An adaptive-hybrid meshfree approximation
method.
Leevan Ling.
International Journal for Numerical Methods in Engineering. 89(5):637-657, 2012Abstract: It is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill-conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a time-evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs.
@ARTICLE{Ling-AdapMeshApprMeth:11, author = {Ling, L.}, title = {An Adaptive--Hybrid Meshfree Approximation Method}, journal = {Int. J. Numer. Methods Eng.}, volume = {89}, number = {5}, pages = {637-657}, year = {2012}, doi = {10.1002/nme.3257} } -
On numerical experiments
for Cauchy problems of elliptic operators.
Fenglian Yang, and Leevan Ling.
Engineering Analysis with Boundary Elements. 35(7):879-882. 2011.Abstract: Over the last decade, there has been a considerable amount of new numerical methods being developed for solving the Cauchy problems of elliptic operators. In this paper, with some new classes of numerical experiments, we re-verify the conclusions in the review article [Wei T, Hon YC, Ling L. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng Anal Bound Elem 2007;31(4):373-85.] concerning the effectiveness of solving Cauchy problems with the method of fundamental solutions.
@ARTICLE{Yang+Ling-numeexpeCaucprob:11, title = {On numerical experiments for {C}auchy problems of elliptic operators}, author = {Yang, F. L. and Ling, L.}, journal = {Eng. Anal. Bound. Elem.}, volume = {35}, number = {7}, pages = {879-882}, year = {2011}, doi = {10.1016/j.enganabound.2011.02.007}, } -
Numerical methods for backward
Markov chain driven Black-Scholes option pricing.
Chi Yan Au, Eric S. Fung, and Leevan Ling.
Frontiers of Mathematics in China. 6(1):17-33, 2011.Abstract:
@ARTICLE{Au+FungETAL-NumemethbackMark:11, title = {Numerical methods for backward {M}arkov chain driven {B}lack-{S}choles option pricing}, author = {Au, C. Y. and Fung, E. S. and Ling, L.}, journal = {Front. Math. China}, volume = {6}, number = {1}, pages = {17-33}, year = {2011}, doi = {10.1007/s11464-010-0089-2}, } -
An energy regularization for
Cauchy problems of Laplace equation in annulus domain.
Houde Han, Leevan Ling, and Tomoya Takeuchi.
Communications in Computational Physics. 9(4):878-896, 2011.Abstract: Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.
@ARTICLE{Han+LingETAL-enerreguCaucprob:11, title = {An energy regularization for {C}auchy problems of {L}aplace equation in annulus domain}, author = {Han, H. and Ling, L. and Takeuchi, T.}, journal = {Commun. Comput. Phys.}, volume = {9}, number = {4}, pages = {878-896}, year = {2011}, doi = {10.4208/cicp.200110.060910a}, } -
Approximate unconditional test
procedure for comparing two ordered multinomials.
Man-Lai Tang, Wai-Yin Poon, Leevan Ling, Yijie Liao, and Hang-Wai Chui.
Computational Statistics and Data Analysis. 55(2): 955-963, 2011.Abstract: The asymptotic and exact conditional methods are widely used to compare two ordered multinomials. The asymptotic method is well known for its good performance when the sample size is sufficiently large. However, Brown et al. (2001) gave a contrary example in which this method performed liberally even when the sample size was large. In practice, when the sample size is moderate, the exact conditional method is a good alternative, but it is often criticised for its conservativeness. Exact unconditional methods are less conservative, but their computational burden usually renders them infeasible in practical applications. To address these issues, we develop an approximate unconditional method in this paper. Its computational burden is successfully alleviated by using an algorithm that is based on polynomial multiplication. Moreover, the proposed method not only corrects the conservativeness of the exact conditional method, but also produces a satisfactory type I error rate. We demonstrate the practicality and applicability of this proposed procedure with two real examples, and simulation studies are conducted to assess its performance. The results of these simulation studies suggest that the proposed procedure outperforms the existing procedures in terms of the type I error rate and power, and is a reliable and attractive method for comparing two ordered multinomials.
@ARTICLE{Tang+PoonETAL-Appruncotestproc:11, title = {Approximate unconditional test procedure for comparing two ordered multinomials}, author = {Tang, M. L. and Poon, W.-L. and Ling, L. and Liao, Y. and Chui, H.-W.}, journal = {Comput. Stats. Data Anal.}, volume = {55}, number = {2}, pages = {955-963}, year = {2011}, doi = {10.1016/j.csda.2010.08.009}, } -
Optimality of the method
of fundamental solutions.
Kwun Ying Wong, and Leevan Ling.
Engineering Analysis with Boundary Elements. 35(1):42-46, 2011.Abstract: The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is also right-hand side vector dependent. The first part of this work, in [EABE 33(5): 637-43], revealed the close connection between the ECN and the accuracy of the Method of Fundamental Solutions (MFS) for each given problem. In this paper, we show how the ECN can help achieve the problem-dependent quasi-optimal settings for MFS calculations—that is, determining the position and density of the source points. A series of examples on Dirichlet and mixed boundary conditions shows the reliability of the proposed scheme; whenever the MFS fails, the corresponding value of the ECN strongly indicates to the user to switch to other numerical methods.
@ARTICLE{Wong+Ling-Optimethfundsolu:11, title = {Optimality of the method of fundamental solutions}, author = {Wong, K. Y. and Ling, L.}, journal = {Eng. Anal. Bound. Elem.}, volume = {35}, number = {1}, pages = {42-46}, year = {2011}, doi = {10.1016/j.enganabound.2010.06.002}, } -
Adaptive-hybrid meshfree method.
Leevan Ling.
In Ch Zhang, M. H. Aliabadi, M. Schanz (Eds.), Advances in Boundary Element Techniques XI, page 252-257. ECltd, England, 2010.Abstract: In talk, we present a hybrid algorithm--making use of both global and compactly supported radial basis functions--for meshfree approximation with minimal fine tunings. The first contribution here is an adaptive node refinement scheme. On the adaptively generated data sites, we apply the global radial basis functions (with adaptive subspace selection) and the compactly supported radial basis functions (with adaptive support selection) that can be used as a blackbox algorithm for robust approximation to a wider class of functions.
@INPROCEEDINGS{Ling-Adapmeshmeth:10, author = {Ling, L.}, editor = {Ch Zhang, M. H. Aliabadi, M. Schanz}, title = {Adaptive-hybrid meshfree method}, booktitle = {Advances in Boundary Element Techniques XI}, year = {2010}, publisher = {ECltd}, address = {England}, pages = {252-257}, } -
Adaptive method of particular
solution for solving 3D inhomogeneous elliptic equations.
C. S. Chen, Ting On Kwok, and Leevan Ling.
International Journal of Computational Methods. 7(3):499-511, 2010.Abstract: Recently, particular solutions using radial basis functions have been used as a basis for solving inhomogeneous partial differential equations as a one-stage approach without the need of finding homogeneous solutions. In this paper, we adopt a newly developed adaptive greedy algorithm to enhance the performance of the one-stage method and alleviate the difficulty of ill-conditioning of the resultant matrix. To demonstrate the effectiveness of coupling these two methods, we give two 3D examples with excellent numerical results.
@ARTICLE{Chen+KwokETAL-Adapmethpartsolu:10, title = {Adaptive method of particular solution for solving {3D} inhomogeneous elliptic equations}, author = {Chen, C. S. and Kwok, T. O. and Ling, L.}, journal = {Int. J. Comput. Methods}, volume = {7}, number = {3}, pages = {499-511}, year = {2010}, doi = {10.1142/S0219876210002271}, } -
An adaptive greedy technique for
inverse boundary determination problem.
Fenglian Yang, Leevan Ling, and Ting Wei.
Journal of Computational Physics. 229(22):8484-8496, 2010.Abstract: In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels.
@ARTICLE{Yang+LingETAL-adapgreetechinve:10, author = {Yang, F. L. and Ling, L. and Wei, T.}, title = {An adaptive greedy technique for inverse boundary determination problem}, journal = {J. Comput. Phys.}, volume = {229}, number = {22}, pages = {8484-8496}, year = {2010}, doi = {10.1016/j.jcp.2010.07.031}, } -
Dimension-splitting data points
redistribution for meshless approximation.
Ting On Kwok, and Leevan Ling.
Journal of Computational and Applied Mathematics. 235(3):736-746, 2010.Abstract: To better approximate nearly singular functions with meshless methods, we propose a data points redistribution method extended from the well-known one-dimensional equidistribution principle. With properly distributed data points, nearly singular functions can be well approximated by linear combinations of global radial basis functions. The proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost. In our numerical examples, clear exponential convergence (with respect to the numbers of data points) can be observed.
@ARTICLE{Kwok+Ling-Dimedatapoinredi:10, author = {Kwok, T. O. and Ling, L.}, title = {Dimension-splitting data points redistribution for meshless approximation}, journal = {J. Comp. Appl. Math.}, volume = {235}, number = {3}, pages = {736-746}, year = {2010}, doi = {10.1016/j.cam.2010.06.026}, } -
Moving-boundary problems
solved by adaptive radial basis functions.
Leopold Vrankar, Edward J. Kansa, Leevan Ling, Franc Runovc, and Goran Turk.
Computers and Fluids. 39(9):1480-1490, 2010.Abstract: The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach.
@ARTICLE{Vrankar+KansaETAL-Moviprobsolvadap:10, author = {Vrankar, L. and Kansa, E. J. and Ling, L. and Runovc, F. and Turk, G.}, title = {Moving-boundary problems solved by adaptive radial basis functions.}, year = {2010}, journal = {Comput. Fluids}, volume = {39}, number = {9}, pages = {1480-1490}, doi = {10.1016/j.compfluid.2010.04.015}, } -
Numerical simulations of 2D
fractional subdiffusion problems.
Hermann Brunner, Leevan Ling, and Masahiro Yamamoto.
Journal of Computational Physics. 229(18):6613-6622, 2010Abstract: The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives. In this work, the 2D fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach. The proposed algorithm is also used to simulate a subdiffusion-convection equation.
@ARTICLE{Brunner+LingETAL-Numesimufracsubd:10, author = {Brunner, H. and Ling, L. and Yamamoto, M.}, title = {Numerical simulations of {2D} fractional subdiffusion problems}, journal = {J. Comput. Phys.}, volume = {229}, number = {18}, pages = {6613-6622}, year = {2010}, doi = {10.1016/j.jcp.2010.05.015}, } -
Confidence intervals for a difference
between proportions based on paired data.
Man-Lai Tang, Man-Ho Ling, Leevan Ling, and Guoliang Tian.
Statistics in Medicine. 29(1): 86-96, 2010.Abstract: We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference between two correlated proportions using the method of variance of estimates recovery (MOVER). The basic idea is to recover variance estimates required for the proportion difference from the confidence limits for single proportions. The CI estimators for a single proportion, which are incorporated with the MOVER, include the Agresti-Coull, the Wilson, and the Jeffreys CIs. Our simulation results show that the MOVER-type CIs based on the continuity corrected Φ coefficient and the Tango score CI perform satisfactory in small sample designs and spare data structures. We illustrate the proposed CIs with several real examples.
@ARTICLE{Tang+LingETAL-Confintediffbetw:10, author = {Tang, M. L. and Ling, M. H. and Ling, L. and Tian, G. L.}, title = {Confidence intervals for a difference between proportions based on paired data}, journal = {Stats. Medicine}, volume = {29}, number = {1}, pages = {86-96}, year = {2010}, doi = {10.1002/sim.3738}, } -
Effective condition
number for boundary knot method.
Fuzhang Wang, Leevan Ling, and Wen Chen.
Computers, Materials & Continua. 12(1):57-70, 2009.Abstract: This study makes the first attempt to apply the effective condition number (ECN) to the stability analysis of the boundary knot method (BKM). We find that the ECN is a superior criterion over the traditional condition number. The main difference between ECN and the traditional condition numbers is that the ECN takes into account the right-hand-side vector to estimate system stability. Numerical results show that the ECN is roughly inversely proportional to the numerical accuracy. Meanwhile, using the effective condition number as an indicator, one can fine-tune the user-defined parameters (without knowledge of the exact solution) to ensure high numerical accuracy from the BKM.
@ARTICLE{Wang+LingETAL-EffeCondNumbBoun:09, author = {Wang, F. Z. and Ling, L. and Chen, W.}, title = {Effective condition number for boundary knot method}, journal = {Comput. Materials Continua}, volume = {12}, number = {1}, pages = {57-70}, year = {2009}, } -
An improved subspace selection algorithm
for meshless collocation methods.
Leevan Ling, and Robert Schaback.
International Journal for Numerical Methods in Engineering. 80(13):1623-1639, 2009.
Abstract: Choosing data points is a common problem for researchers who employ various meshless methods for solving partial differential equations. On the one hand, high accuracy is always desired; on the other, ill-conditioning problems of the resultant matrices, which may lead to unstable algorithms, prevent some researchers from using meshless methods. For example, the optimal placements of source points in the method of fundamental solutions or of the centers in the radial basis functions method are always unclear. Intuitively, such optimal locations will depend on many factors: the partial differential equations, the domain, the trial basis used (i.e. the employed method itself), the computational precisions, some user-defined parameters, and so on. Such complexity makes the hope of having an optimal centers placement unpromising. In this paper, we provide a data-dependent algorithm that adaptively selects centers based on all the other variables.
@ARTICLE{Ling+Schaback-imprsubsselealgo:09, author = {Ling, L. and Schaback, R.}, title = {An improved subspace selection algorithm for meshless collocation methods}, journal = {Int. J. Numer. Methods Eng.}, volume = {80}, number = {13}, pages = {1623-1639}, year = {2009}, doi = {10.1002/nme.2674}, } -
On convergence of a
least-squares Kansa's method for the modified Helmholtz equations.
Ting On Kwok, and Leevan Ling.
Advances in Applied Mathematics and Mechanics. 1(3):367-382, 2009.Abstract:
@ARTICLE{Kwok+Ling-convleasKansmeth:09, author = {Kwok, T. O. and Ling, L.}, title = {On convergence of a least-squares {K}ansa's method for the modified {H}elmholtz equations}, year = {2009}, journal = {Adv. Appl. Math. Mech.}, volume = {1}, number = {3}, pages = {367--382}, } -
Numerical simulation of
two-dimensional combustion using mesh-free methods.
Edward J. Kansa, Ralph C. Aldredge, and Leevan Ling.
Engineering Analysis with Boundary Elements. 33(7):940-950, 2009.Abstract: The purpose of this research was to develop tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs). Mesh-free methods offer many distinct advantages over traditional finite difference, finite element, and finite volume methods. Traditional Lagrangian methods with significant swirl require mesh stiffeners and periodic remeshing to avoid excessive mesh distortion; such codes often require user interaction to repair the meshes before the simulation can proceed again. A propagating flame of infinite extent is simulated as a collection of normalized cells with periodic boundary conditions. Rather than capturing the flame front, it is tracked as a discontinuity. The flame front is approximated as a product of a Heaviside function in the normal propagation direction and a piece-wise continuous function represented by RBFs in the tangential direction. The cells are subdivided into the burned and unburned sub-domains approximated by two-dimensional periodic RBFs that are constrained to be strictly conservative. The underlying steady flow is vortical with an input turbulent intensity. The governing equations are rotationally and translationally transformed to produce exact differentials that are integrated exactly in time. In the present paper, the previous results of Aldredge who used a finite-difference level-set method were compared. The physical behavior was remarkably similar, whereas the finite-difference level-set method required 14 h of CPU time, the RBF approach required only 120 CPU seconds on a desktop computer for the case with the largest turbulent intensity. Although there are no other papers that tried to duplicate the original results of Aldredge, the results that are reported here are consistent with the physics observed in other experimental and numerical investigations.
@ARTICLE{Kansa+AldredgeETAL-Numesimutwo-comb:09, author = {Kansa, E. J. and Aldredge, R. C. and Ling, L.}, title = {Numerical simulation of two-dimensional combustion using mesh-free methods}, journal = {Eng. Anal. Bound. Elem.}, volume = {33}, number = {7}, pages = {940-950}, year = {2009}, doi = {10.1016/j.enganabound.2009.02.008}, } -
Point sources identification
problems for heat equations.
Leevan Ling, and Tomoya Takeuchi.
Communications in Computational Physics. 5(5):897-913, 2009.Abstract: We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points. We aim to identify the unknown number of sources and their locations along with their strengths. In our previous work, we proved that minimum measurement points needed under the noise-free setting. In this paper, we extend the proof to cover the noisy cases over a border class of source functions. We show that if the regularization parameter is chosen properly, the problem can be transformed into a poles identification problem. A reconstruction scheme is proposed on the basis of the developed theoretical results. Numerical demonstrations in 2D and 3D conclude the paper.
@ARTICLE{Ling+Takeuchi-PoinSourIdenProb:09, author = {Ling, L. and Takeuchi, T.}, title = {Point sources identification problems for heat equations}, journal = {Commun. Comput. Phys.}, volume = {5}, number = {5}, pages = {897-913}, year = {2009}, doi = {10.4208/cicp.2009.v5.p897} } -
On convergent numerical
algorithms for unsymmetric collocation.
Cheng-Feng Lee, Leevan Ling, and Robert Schaback.
Advances in Computational Mathematics. 30(4):339-354, 2009.Abstract: In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa's method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa's method are examined and verified in arbitrary-precision computations. Numerical examples confirm with the theories that the modified Kansa's method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa's method. In double-precision, even for very large RBF shape parameters, we show that the modified Kansa's method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.
@ARTICLE{Lee+LingETAL-convnumealgounsy:09, author = {Lee, C. F. and Ling, L. and Schaback, R.}, title = {On convergent numerical algorithms for unsymmetric collocation}, journal = {Adv. Comput. Math}, volume = {30}, number = {4}, pages = {339-354}, year = {2009}, doi = {10.1007/s10444-008-9071-x}, } -
Applicability of the
method of fundamental solutions.
Tyler W. Drombosky, Ashley L. Meyer, and Leevan Ling.
Engineering Analysis with Boundary Elements. 33(5):637-643, 2009.Abstract: The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number. Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.
@ARTICLE{Drombosky+MeyerETAL-Applmethfundsolu:09, author = {Drombosky, T. W. and Meyer, A. L. and Ling, L.}, title = {Applicability of the method of fundamental solutions}, journal = {Eng. Anal. Bound. Elem.}, volume = {33}, number = {5}, pages = {637-643}, year = {2009}, } -
Boundary control for
inverse Cauchy problems of the Laplace equations.
Leevan Ling, and Tomoya Takeuchi.
Computer Modeling in Engineering & Sciences. 29(1):45-54, 2008.Abstract: The method of fundamental solutions is coupled with the boundary control technique to solve the Cauchy problems of the Laplace equations. The main idea of the proposed method is to solve a sequence of direct problems instead of solving the inverse problem directly. In particular, we use a boundary control technique to obtain an approximation of the missing Dirichlet boundary data; the Tikhonov regularization technique and the $L$-curve method are employed to achieve such a goal stably. Once the boundary data on the whole boundary are known, the numerical solution to the Cauchy problem can be obtained by solving a direct problem. Numerical examples are provided for verifications of the proposed method on the steady-state heat conduction problems.
@ARTICLE{Ling+Takeuchi-BouncontinveCauc:08, author = {Ling, L. and Takeuchi, T.}, title = {Boundary control for inverse {C}auchy problems of the {L}aplace equations}, journal = {Comput. Model. Eng. Sci.}, volume = {29}, number = {1}, pages = {45-54}, year = {2008}, } -
Stable and convergent unsymmetric
meshless collocation methods.
Leevan Ling, and Robert Schaback.
SIAM Journal on Numerical Analysis. 46(3):1097-1115, 2008.Abstract: In the theoretical part of this paper, we introduce a simplified proof technique for error bounds and convergence of a variation of Kansa's well-known unsymmetric meshless collocation method. For a numerical implementation of the convergent variation, a previously proposed greedy technique is coupled with linear optimization. This algorithm allows a fully adaptive on-the-fly data-dependent meshless selection of test and trial spaces. The new method satisfies the assumptions of the background theory, and numerical experiments demonstrate its stability.
@ARTICLE{Ling+Schaback-Stabconvunsymesh:08, author = {Ling, L. and Schaback, R.}, title = {Stable and convergent unsymmetric meshless collocation methods}, journal = {SIAM J. Numer. Anal.}, volume = {46}, number = {3}, pages = {1097-1115}, year = {2008}, doi = {10.1137/06067300X}, } -
Arbitrary
precision computations of Kansa's method.
Leevan Ling.
In A. Ferreira, Edward J. Kansa, G. Fasshauer, V. Leitão (Eds.), Progress on meshless methods, 11, 77-83, Springer, New York, 2008.Abstract: In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa's method. The rates of convergence of two variations of Kansa's method are examined and verified in arbitrary—precision computations.
@INPROCEEDINGS{Ling-ArbipreccompKans:08, author = {Ling, L.}, editor = {Ferreira, A. and Kansa, E. J. and Fasshauer, G. and Leit\~ao, V.}, title = {Arbitrary precision computations of Kansa's method}, booktitle = {Progress on meshless methods}, year = {2008}, publisher = {Springer}, address = {New York}, volume = {11}, pages = {77-83}, } -
Stability analysis for
the penalty plus hybrid and the direct Trefftz methods for singularity problems.
Zi-Cai Li, Hung-Tsai Huang, Jin Huang, and Leevan Ling.
Engineering Analysis with Boundary Elements. 31(2):163-175, 2007.Abstract: For solving the linear algebraic equations $Ax=b$, the new stability analysis is made based on the effective condition number $Cond_{\mathrm{eff}}$. The $Cond_{\mathrm{eff}}$ may provide a better upper bound of relative errors of $x$ resulting from the rounding errors of $b$, than the traditional condition number $Cond$, which with too large value is, in many times, misleading. In this paper, we apply the effective condition number to the Trefftz methods (TMs) for Poisson's equations with singularities. Two TMs, such as the penalty plus hybrid TM and the Lagrange multiple (i.e., direct) TM, are studied. We focus on the stability analysis of the solutions when the optimal superconvergence is achieved. When solving Motz's problem, the benchmark of singularity problems, by the penalty plus hybrid TM, we have derived that $Cond_{\mathrm{eff}} = O(N(\sqrt{2})^N)$ and $Cond = O(N^2 2^N)$, where $N$ is the number of the singular particular functions used. For solving Motz's problem by the direct TM, we have derived that $Cond/Cond_{\mathrm{eff}} = O(N\sqrt{2}^{\,N})$. Numerical experiments are provided to verify the stability analysis made. In summary, the two TMs are efficient, but the penalty plus hybrid TM is more recommended, due to simplicity of algorithms without extra-variables and less limitations in applications.
@ARTICLE{Li+HuangETAL-Stabanalpenaplus:07, author = {Li, Z. C. and Huang, H. T. and Huang, J. and Ling, L.}, title = {Stability analysis for the penalty plus hybrid and the direct {T}refftz methods for singularity problems}, journal = {Eng. Anal. Bound. Elem.}, volume = {31}, number = {2}, pages = {163-175}, year = {2007}, doi = {10.1016/j.enganabound.2006.04.005}, } -
Method of fundamental solutions with regularization techniques for Cauchy problems of
elliptic operators.
Ting Wei, Yiu Chung Hon, and Leevan Ling.
Engineering Analysis with Boundary Elements. 31(4): 373-385, 2007.Abstract: In this paper we combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and $L$-curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.
@ARTICLE{Wei+HonETAL-Methfundsoluwith:07, author = {Wei, T. and Hon, Y. C. and Ling, L.}, title = {Method of fundamental solutions with regularization techniques for {C}auchy problems of elliptic operators}, journal = {Eng. Anal. Bound. Elem.}, volume = {31}, number = {4}, pages = {373-385}, year = {2007}, doi = {10.1016/j.enganabound.2006.07.010}, } -
An accurate refinement
scheme for inverse heat source location identifications.
Leevan Ling, and Tomoya Takeuchi.
Computer Modeling in Engineering & Sciences. 20(2): 99-110. 2007.Abstract: We aim to identify the unknown source locations in a two-dimensional heat equation from scattered measurements. In [Inverse Problems, 22(4):1289--1305, 2006], we proposed a numerical procedure that identifies the unknown source locations of a 2D heat equation solely based on three measurement points. Due to the nonlinearity and complexity of the problem, the quality of the resulting estimations is often poor especially when the number of unknowns is large. In this paper, we propose a linear refinement scheme that takes the outputs of the existing nonlinear algorithm as initial guesses and iteratively improves on the accuracy of the estimations; the convergence of the proposed algorithm with noisy data is proven. The work is concluded by some numerical examples.
@ARTICLE{Ling+TakeuchiETAL-accurefischeinve:07, author = {Ling, L. and Takeuchi, T.}, title = {An accurate refinement scheme for inverse heat source location identifications}, journal = {Comput. Model. Eng. Sci.}, volume = {20}, number = {2}, pages = {99-110}, year = {2007}, } -
A computational method for solving Cauchy problems of elliptic operators.
Yiu Chung Hon, Ting Wei, and Leevan Ling.
In G. R. Liu, V. B. C. Tan, and X. Han (Eds), Computational methods, 1375-1384. Springer, New York, 2006.Abstract: In this talk we combine a meshless method of fundamental solution (MFS) with different regularization methods to solve Cauchy problems of elliptic differential operators. The main idea of MFS is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the computational domain. Three regularization methods based on the singular value decomposition with four different choice strategies for regularization parameters are proposed.
@INPROCEEDINGS{Hon+Ling-compmethsolvCauc:06, author = {Hon, Y. C. and Wei, T. and Ling, L.}, editor = {Liu, G. R. and Tan, V. B. C. and Han, X. }, title = {A computational method for solving Cauchy problems of elliptic operators}, booktitle = {Computational methods}, year = {2006}, publisher = {Springer}, address = {New York}, pages = {1375-1384}, } -
Identification of source locations in two-dimensional heat equations.
Leevan Ling, Yiu Chung Hon, Masahiro Yamamoto, and T. Takeuchi.
Inverse Problems. 22(4):1289-1305, 2006.Abstract: In this paper, we show the uniqueness of the identification of unknown source locations in two-dimensional heat equations from scattered measurements. Based on the assumption that the unknown source function is a sum of some known functions, we prove that one measurement point is sufficient to identify the number of sources and three measurement points are sufficient to determine all unknown source locations. For verification, we propose a numerical reconstruction scheme for recovering the number of unknown sources and all source locations.
@ARTICLE{Ling+YamamotoETAL-Idensourlocatwo-:06, author = {Ling, L. and Yamamoto, M. and Hon, Y. C. and Takeuchi, T.}, title = {Identification of source locations in two-dimensional heat equations}, journal = {Inverse Probl.}, volume = {22}, number = {4}, pages = {1289-1305}, year = {2006}, doi = {10.1088/0266-5611/22/4/011}, } -
Finding numerical derivatives for unstructured and noisy data by multiscale kernels.
Leevan Ling.
SIAM Journal on Numerical Analysis. 44(4):1780-1800, 2006.Abstract: The recently developed multiscale kernel of R. Opfer [Adv. Comput. Math., 25 (2006), pp. 357-380] is applied to approximate numerical derivatives. The proposed method is truly mesh-free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L-curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.
@ARTICLE{Ling-Findnumederiunst:06, author = {Ling, L.}, title = {Finding numerical derivatives for unstructured and noisy data by multiscale kernels}, journal = {SIAM J. Numer. Anal.}, volume = {44}, number = {4}, pages = {1780-1800}, year = {2006}, doi = {10.1137/050630246}, } -
Adaptive multiquadric
collocation for boundary layer problems.
Leevan Ling, and Manfred R. Trummer.
Computational and Applied Mathematics. 188(2):265-282, 2006.Abstract: An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution.
@ARTICLE{Ling+Trummer-Adapmultcollboun:06, author = {Ling, L. and Trummer, M. R.}, title = {Adaptive multiquadric collocation for boundary layer problems}, journal = {J. Comp. Appl. Math.}, volume = {188}, number = {2}, pages = {265-282}, year = {2006}, doi = {10.1016/j.cam.2005.04.018}, } -
Results on meshless
collocation techniques.
Leevan Ling, Roland Opfer, and Robert Schaback.
Engineering Analysis with Boundary Elements.30(4): 247-253, 2006.Abstract: Though the technique introduced by Kansa is very successful in engineering applications, there were no proven results so far on the unsymmetric meshless collocation method for solving PDE boundary value problems in strong form. While the original method cannot be proven to be fail-safe in general, we prove asymptotic feasibility for a generalized variant using separated trial and test spaces. Furthermore, a greedy variation of this technique is provided, allowing a fully adaptive matrix-free and data-dependent meshless selection of the test and trial spaces.
@ARTICLE{Ling+OpferETAL-Resumeshcolltech:06, author = {Ling, L. and Opfer, R. and Schaback, R.}, title = {Results on meshless collocation techniques}, journal = {Eng. Anal. Bound. Elem.}, volume = {30}, number = {4}, pages = {247-253}, year = {2006}, doi = {10.1016/j.enganabound.2005.08.008}, } -
The role of the multiquadric
shape parameters in solving elliptic partial differential equations.
J. Wertz, Edward J. Kansa, and Leevan Ling.
Computers and Mathematics with Applications. 51(8):1335-1348, 2006.Abstract: This study examines the generalized multiquadrics (MQ), $\phi_j(x) = [(x-x_j)^2+c_j^2]^\beta$, in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent $\beta$ as well as $c_j^2$ can be classified as shape parameters since these affect the shape of the MQ basis function. We examine variations of $\beta$ as well as $c_j^2$, where $c_j^2$ can be different over the interior and on the boundary. The results show that increasing $\beta$ has the most important effect on convergence, followed next by distinct sets of $(c_j^2)_{\Omega\setminus\partial\Omega} \ll (c_j^2)_{\partial\Omega}$. Additional convergence accelerations are obtained by permitting both $(c_j^2)_{\Omega\setminus\partial\Omega}$ and $(c_j^2)_{\partial\Omega}$ to oscillate about their mean value with amplitude of approximately $1/2$ for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.
@ARTICLE{Wertz+KansaETAL-rolemultshappara:06, author = {Wertz, J. and Kansa, E. J. and Ling, L.}, title = {The role of the multiquadric shape parameters in solving elliptic partial differential equations}, journal = {Comput. Math. Appl.}, volume = {51}, number = {8}, pages = {1335-1348}, year = {2006}, doi = {10.1016/j.camwa.2006.04.009}, } -
Inverse source identification for
Poisson equation.
Leevan Ling, Yiu Chung Hon, and Masahiro Yamamoto.
Inverse Problems in Science and Engineering. 13(4):433-447, 2005.Abstract: A numerical method for identifying the unknown point sources for a two-dimensional Poisson problem from Dirichlet boundary data is proposed. Under an assumption that the total number and estimate positions of the point sources are known, the exact positions and corresponding strengths of the distinct point sources can be identified from scattered (noisy) observed Dirichlet boundary data. Numerical verification indicated that the method is efficient and robust.
@ARTICLE{Ling+HonETAL-InvesouridenPois:05, author = {Ling, L. and Hon, Y. C. and Yamamoto, M.}, title = {Inverse source identification for {P}oisson equation}, journal = {Inverse Probl. Sci. Eng.}, volume = {13}, number = {4}, pages = {433-447}, year = {2005}, doi = {10.1080/17415970500126500}, } -
On approximate cardinal
preconditioning methods for solving PDEs with radial basis functions.
Damian Brown, Leevan Ling, Edward J. Kansa, and Jeremy Levesley.
Engineering Analysis with Boundary Elements. 29(4):343-353, 2005.Abstract: The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.
@ARTICLE{Brown+LingETAL-apprcardprecmeth:05, author = {Brown, D. and Ling, L. and Kansa, E. and Levesley, J.}, title = {On approximate cardinal preconditioning methods for solving {PDE}s with radial basis functions}, journal = {Eng. Anal. Bound. Elem.}, volume = {29}, number = {4}, pages = {343-353}, year = {2005}, doi = {10.1016/j.enganabound.2004.05.006}, } -
Numerical analysis of parameters in
a laminated beam model by radial basis functions.
Yiu Chung Hon, Leevan Ling, and K. M. Liew.
Computers, Materials & Continua. 2(1):39-50, 2005.Abstract: In this paper we investigate a thermally driven micro-electrical-mechanical system which was originally designed for an inkjet printer to precisely deliver small ink droplets onto paper. In the model, a tiny free-ended beam of metal bends and projects ink onto paper. The model is solved by using the recently developed radial basis functions method. We establish the accuracy of the proposed approach by comparing the numerical results with reported experimental data. Numerical simulations indicate that a light (low composite mass) beam is more stable as it does not oscillate much. A soft (low rigidity) beam results in a higher rate of deflection, when compared to a high rigidity one. Effects caused by the values of physical parameters are also studied. Finally, we give a prediction on the optimal time for the second current pulse which results in maximum rate of second deflection of the beam.
@ARTICLE{Hon+LingETAL-Numeanalparalami:05, author = {Hon, Y. C. and Ling, L. and Liew, K. M.}, title = {Numerical analysis of parameters in a laminated beam model by radial basis functions}, journal = {Comput. Materials Continua}, volume = {2}, number = {1}, pages = {39-50}, year = {2005}, doi = {10.3970/cmc.2005.002.039} } -
A least squares preconditioner
for radial basis functions collocation methods.
Leevan Ling, and Edward J. Kansa.
Advances in Computational Mathematics. 23(1-2):31-54, 2005.Abstract: Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring $O(N^3)$ flops. We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements. The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain. Our preconditioner requires $O(mN^2)$ flops to set up, and $O(mN)$ storage locations where $m$ is a user define parameter of order of 10. For the 2D MQ-RBF with the shape parameter $c \sim 1/\sqrt{N}$, the number of iterations required for convergence is of order of 10 for large values of $N$, making this a very attractive approach computationally. As the shape parameter increases, our preconditioner will eventually be affected by the ill conditioning and round-off errors, and thus becomes less effective. We tested our preconditioners on increasingly larger $c$ and $N$. A more stable construction scheme is available with a higher set up cost.
@ARTICLE{Ling+Kansa-leasprecradibasi:05, author = {Ling, L. and Kansa, E. J.}, title = {A least-squares preconditioner for radial basis functions collocation methods}, journal = {Adv. Comput. Math.}, volume = {23}, number = {1-2}, pages = {31-54}, year = {2005}, doi = {10.1007/s10444-004-1809-5}, } -
Multivariate quasi-interpolation
formula with dimension-splitting multiquadric basis.
Leevan Ling.
Applied Mathematics and Computation. 161(1):195-209, 2005.Abstract: In this paper, we extend the multilevel univariate quasi-interpolation formula proposed in [A univariate quasi-multiquadric interpolation with better smoothness, Comput. Math. Appl., in press] to multidimensions using the dimension-splitting multiquadric (DSMQ) basis function approach. Our multivariate scheme is readily performed on parallel computers. We show that the cost of finding the coefficient of the quasi-interpolant is $3dN$ on $\mathbb{R}^d$, and the work of direct evaluation of the quasi-interpolant can be reduced from $11N^2$ in 2D and $16N^2$ in 3D to approximately $2N$. A boundary padding technique can be employed to improve accuracy. Numerical results in 2D and 3D are both given.
@ARTICLE{Ling-Multquasscheford:05, author = {Ling, L.}, title = {Multivariate quasi-interpolation schemes for dimension-splitting multiquadric}, journal = {Appl. Math. Comput.}, volume = {161}, number = {1}, pages = {195-209}, year = {2005}, doi = {10.1016/j.amc.2003.12.022}, } -
Improved numerical
solver for Kansa's method based on affine space decomposition.
Leevan Ling, and Yiu Chung Hon.
Engineering Analysis with Boundary Elements. 29(12):1077-1085, 2005.Abstract: Radial Basis functions (RBFs) have been successfully developed as a truly mesh-free method to find the numerical solutions of partial differential equations (PDEs). In particular, the asymmetric RBF collocation method (Kansa's method) is one of the most frequently used methods due to its ease of implementation. To achieve high accuracy, the resultant system of RBF–PDE problem usually becomes badly conditioned. We propose in this paper an improved solution method based on an affine space decomposition that decouples the influence between the interior and boundary collocations. Numerical examples are given to compare the proposed method with several direct methods.
@ARTICLE{Ling+Hon-ImprnumesolvKans:05, author = {Ling, L. and Hon, Y. C.}, title = {Improved numerical solver for {K}ansa's method based on affine space decomposition}, journal = {Eng. Anal. Bound. Elem.}, volume = {29}, number = {12}, pages = {1077-1085}, year = {2005}, doi = {10.1016/j.enganabound.2005.07.003}, } -
Multiquadric collocation
method with integral formulation for boundary layer problems.
Leevan Ling, and Manfred R. Trummer.
Computers and Mathematics with Applications. 48(5-6):927-941, 2004.Abstract: Singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly. This paper concentrates on the case where theses layers occur near the boundary, although our method can be applied to problems with interior layers. One technique to deal with the increased resolution requirements in these layers is the use of domain transformations. A coordinate stretching based transform allows to move collocation points into the layer, a requirement to resolve the layer accurately. Previously, such transformations have been studied in the context of finite-difference and spectral collocation methods. In this paper, we use radial basis functions (RBFs) to solve the boundary value problem. Specifically, we present a collocation method based on multiquadric (MQ) functions with an integral formulation combined with a coordinate transformation. We find that our scheme is ultimately more accurate than a recently proposed adaptive MQ scheme. The RBF scheme is also amenable to adaptivity.
@ARTICLE{Ling+Trummer-Multcollmethwith:04, author = {Ling, L. and Trummer, M. R.}, title = {Multiquadric collocation method with integral formulation for boundary layer problems}, journal = {Comput. Math. Appl.}, volume = {48}, number = {5-6}, pages = {927-941}, year = {2004}, doi = {10.1016/j.camwa.2003.06.010}, } -
A univariate quasi
multiquadric interpolation with better smoothness.
Leevan Ling.
Computers and Mathematics with Applications. 48(5-6):897-912, 2004.Abstract: In this paper, we propose a multilevel univariate quasi-interpolation scheme using a multiquadric basis. It is practical as it does not require derivative values of the function being interpolated. It has a higher degree of smoothness than the original level-0 formula as it allows a shape parameter $c=O(h)$. Our level-1 quasi-interpolation costs $O(n\log n)$ flops to set up. It preserves strict convexity and monotonicity. When $c=O(h)$, we prove the proposed scheme converges with a rate of $O(h^{2.5}\log h)$. Furthermore, if both $|f''(a)|$ and $|f''|$ are relatively small compared with $|f''|_\infty$, the convergence rate will increase. We verify numerically that $c=h$ is a good shape parameter to use for our method; hence, we need not find the optimal parameter. For all test functions, both convergence speed and error are optimized for $c$ between $0.5h$ and $1.5h$. Our method can be generalized to a multilevel scheme; we include the numerical results for the level-2 scheme. The shape parameter of the level-2 scheme can be chosen between $2h$ and $3h$.
@ARTICLE{Ling-univquasintewith:04, author = {Ling, L.}, title = {A univariate quasi-multiquadric interpolation with better smoothness}, journal = {Comput. Math. Appl.}, volume = {48}, number = {5-6}, pages = {897-912}, year = {2004}, doi = {10.1016/j.camwa.2003.05.014}, } -
Preconditioning for radial basis
functions with domain decomposition methods.
Mathematical and Computer Modelling. 40(13):1413-1427, 2004.Abstract: In our previous work, an effective preconditioning scheme based on constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements showed very attractive numerical results. The preconditioner costs $O(N^2)$ flops to set up and $O(N)$ storage. The preconditioning technique is sufficiently general that it can be applied to different types of differential operators. It was applied to the 2D multiquadric method, with $c\sim 1/\sqrt{N}$, on the Poisson test problem, and the preconditioned GMRES converges in a small number of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We study different implementations of the ACBF-DDM scheme and provide numerical results for $N>10{,}000$ nodes. We demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.
@ARTICLE{Ling+Kansa-Precradibasifunc:04, author = {Ling, L. and Kansa, E. J.}, title = {Preconditioning for radial basis functions with domain decomposition methods}, journal = {Math. Comput. Modelling}, volume = {40}, number = {13}, pages = {1413-1427}, year = {2004}, doi = {10.1016/j.mcm.2005.01.002}, } -
A volumetric integral
radial basis function method for time dependent partial differential equations.
Edward J. Kansa, Henry Power, Gregory E. Fasshauer, and Leevan Ling.
Engineering Analysis with Boundary Elements. 28(10):1191-1206, 2004.Abstract: A strictly conservative volume-integral formulation of the time-dependent conservation equations in terms of meshless radial basis functions (RBFs) is presented. Rotational and translational transformations are considered that simplify the partial differential equations (PDEs) to be solved. As a result, the solutions represented at a finite set of knots, $x\in\Omega\subset\mathbb{R}^d$, are permitted to move as the system of equations evolves in time. Knots are inserted, deleted, or rearranged in such a manner as to conserve the extensive physical quantities of mass, momentum components, and total energy. Our study consists of the following parts: (A) Local rotational and Galilean translational transformations can be obtained to reduce the conservation equations to steady-state forms for the inviscid Euler equations or Navier--Stokes equations. (B) The entire set of PDEs is transformed into the method-of-lines approach, yielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time. (C) The spatial components are approximated by expansions of meshless RBFs; each individual RBF is volumetrically integrated at one of the sampling knots $x_i$, yielding a collocation formulation of the method-of-lines structure of the ODEs. (D) Because the volume-integrated RBFs increase more rapidly away from the data center than the commonly used RBFs, we use a higher-order preconditioner to counteract the ill-conditioning problem. Domain decomposition is used over each piecewise continuous subdomain.
@ARTICLE{Kansa+PowerETAL-voluinteradibasi:04, author = {Kansa, E. J. and Power, H. and Fasshauer, G. E. and Ling, L.}, title = {A volumetric integral radial basis function method for time-dependent partial differential equations. {I}. Formulation}, journal = {Eng. Anal. Bound. Elem.}, volume = {28}, number = {10}, pages = {1191-1206}, year = {2004}, doi = {10.1016/j.enganabound.2004.01.004}, }
Edited Special Issues:
-
Advanced Mesh-based and
Particle-based Numerical Methods for Engineering and Applied Mathematics Problems.
C.-T. Wu, L. Wang, B. Bonello, Leevan Ling, N. Ma, M. A. Schweitzer (Eds.), Mathematical Problems in Engineering, Hindawi, 2017 -
Special Issue on Inverse
Problems and Applications.
Masahiro Yamamoto, J. Cheng, Yiu Chung Hon, J.-Y. Lee, L. Ling, G.-R. Liu, J. Wang (Eds.), Applicable Analysis, 91(4), 2012. -
Proceeding of the International
Conference on Inverse Problems 2010.
Yiu Chung Hon, and Leevan Ling (Eds), Journal of Physics: Conference Series, Vol. 290, 2011. -
Numerical
methods and analysis for PDEs and inverse problems.
C. S. Chen, M.-C. Lai, Leevan Ling, J. Li, M. Yamamoto (Eds.), Advances in Applied Mathematics and Mechanics, 1(6), 2009.
Textbooks:
-
Mathematics of Fairness.
S. N. Chiu, and Leevan Ling.
Series: Hong Kong Mathematical Society Texts in General Education, Vol. 1, Pearson Education Asia, 2012. -
Manage Your Money without Formulas.
Leevan Ling.
Series: Hong Kong Mathematical Society Texts in General Education, Vol. 2, Pearson Education Asia, 2012. -
Discovering Hong Kong by SPSS.
H. W. Chiu, and Leevan Ling.
Series: Hong Kong Mathematical Society Texts in General Education, Vol. 3, Pearson Education Asia, 2012. -
Estimating the World.
W. S. Don, and Leevan Ling.
Series: Hong Kong Mathematical Society Texts in General Education, Vol. 4, Pearson Education Asia, 2012. -
Smart Decisions.
W. C. Shiu, and Leevan Ling.
Series: Hong Kong Mathematical Society Texts in General Education, Vol. 8, Pearson Education Asia, 2012.
