Research

Members, visitors, postdoctoral fellows and students affiliated with the JRIAM are engaged in frontier research in a variety of areas. These include:

• adaptive and spectral methods for numerical computation
• applied probability
• computational finance
• computational fluid dynamics
• data mining
• experimental design
• graph theory
• high dimensional integration and approximation
• Monte Carlo methods
• parallel computing
• optimization

Most of these research areas involve more than one member of the JRIAM, and many members of the JRIAM are active in more than one area. For more details check the websites of the members of the JRIAM.

Featured Research Programmes

The JRIAM conducts research in a number of fields in applied mathematics. Some of the ongoing research programmes are described below. We have aimed to build cohesive research teams. Each research area involves two or more members of the JRIAM plus visitors, postdoctoral fellows, and students. The references for the cited publications appear in section Publications.

Experimental design

In order to obtain as much information as possible from an experiment it must be well designed. That is, for each experiment the levels of the factors one can control must be chosen carefully. The uniform design, of which Prof. Kai-Tai Fang is one of the pioneers, spreads design points as evenly as possible over the experimental domain. Recent research by the JRIAM in this area has included demonstrating the efficiency and robustness of the uniform design and showing how uniform design improve upon traditional fractional factorial designs by increasing the number of levels while maintaining an even spread of design points [Fan02, FMM02, HL02, MFL03]. New algorithms have been developed to construct uniform designs [FQ02, FLQ03]. Researchers involved: K.-T. Fang, F. J. Hickernell, M. Y. Ai, D. K. J. Lin, R. Mukerjee, A. Zhang, Y. Zhang

High dimensional integration and approximation

Integrals of multivariate functions arise in a number of applications in finance, physics, statistics and other fields. Pricing financial derivatives is one example. Typically these integrals are approximated by weighted averages of integrand values at certain well-chosen points. A related problem is how to pproximate a function of many variables based on function values at certain points. This research effort involves discovering new algorithms to solve such problems, analyzing the error of algorithms, and determining when the convergence rates can be made nearly independent of the dimension of the problem. Recent research results have extended the error analysis to the important case where the integration domain is unbounded (as in finance) [HSW03, HSW04a, HSW04b, HSW04c, HSW04d]. Work has also been done on expanding the spaces of functions for which a given convergence rate holds [FH03a, FH03b, HF03]. Researchers involved: F. J. H ickernell, G. Fang, F. Huang, G. Wasilkowski, G. Wei, X. Y. Zang

Moving mesh methods for partial differential equations

Adaptive mesh methods have important applications for a variety of physical and engineering areas such as solid and fluid dynamics, combustion, heat transfer, material science etc. The physical phenomena in these areas develop dynamically singular or nearly singular solutions in fairly localized regions, such as shock waves, boundary layers, detonation waves etc. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost. Recent results include solving incompressible flow problems with moving finite element methods [DLTZ03]. This work presents the first effort in designing moving mesh algorithm to solve the incompressible Navier-Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep the divergence-free for the velocity field at each time level. The proposed numerical scheme extends a recent adaptive grid method based on harmonic mapping [LTZ01, LTZ02]. Researchers involved: T. Tang. P.-W. Zhang, R. Li, Y.-N. Di, Z.-R. Zhang, W.-L. Lee

Numerical solutions for PDEs in large solution domains

Many physical problems are posed in large solution domains. To make numerical computations possible, the solution domain has to be truncated into a finite region, and therefore artificial boundary conditions have to be designed. Moreover, even with successful domain truncation, fast solution solvers are extremely important for the underlying PDE problems in large solution domains. Recent results in this area include new algorithms for problems in solid mechanics [WX03] and for problems in computational finance [HW03]. Researchers involved: T. Tang, X.-N. Wu, H.-D. Han, Z.-H. Teng, J.-C. Jin

High accuracy methods

Spectral methods are highly accurate methods for solving partial differential equations and similar problems. Radial basis function methods, allow more freedom in the placement of design points, but have recently been shown to also obtain the high accuracy of spectral methods. Research by JRIAM members includes developing Hermite spectral method for partial differential equations in unbounded domains [FGT02] and a multi-grid method for elliptical equations [HSTX03]. There has also been recent success in extending spectral methods to high dimensional problems by sampling on lattices [LH03a,LH03b]. Researchers involved: F. J. Hickernell, T. Tang, W. M. Xue, B. Fornberg, D. Li

Statistical inference

An important part of statistics is the inference of parameter values from data. For more complicated problems finding estimators that are consistent, efficient and robust is a difficult problem. Recent results by [HYFW02] develop consistent estimates for the intensity of droplets from a spray flame when the recording instrument is subject to dead time. Other recent work in statistical inference includes [MHF04, PF02]. Researchers involved: K.-T. Fang, S.-Y. He, C. L. Mei

External Grants:

• Liao, Li-Zhi, HKBU12319816, Sept. 1, 2016 -- Aug. 31, 2019, HK$326,811, ``Neurodynamical approach for linearly constrained convex programming’’
• Liao, Li-Zhi, HKBU12302019, Sept. 1, 2019 -- Aug. 31, 2022, HK$332,261, ``Interior point polynomial-time algorithms for linearly constrained convex programming’’
• Liao, Li-Zhi, HKBU12300920, Jan. 1, 2021 -- June30, 2023, HK$396,967, ``Neurodynamical approach for optimization problems with partial orthogonality constraints in supervised learning’’
• Cheng, Ming-Yen, Essence codings in functional linear regression: identification, interpretation and application (HKBU12304120), RGC General Research Fund, HK$ 599,861, January 2021-December 2023.
• Ling, Leevan, GRF/12301520 [11.23.4551.173072] $576,337, Kernel-based approximation methods for curvatures estimations and applications on raw point clouds, 1/2021-12/2023.
• Ling, Leevan, GRF/12301419 [11.23.4551.172940] $332,261, On the Asymptotics of Least-squares Kernel Collocation Methods, 1/2020-12/2022.
• Ling, Leevan, GRF/12303818 [11.23.4551.172846] $456,452, Meshfree Methods for Coupled Bulk-Surface Problems, 1/2019-12/2021.
• Ling, Leevan, GRF/12301917 [11.23.4551.172746] $314,900, Meshfree methods for parabolic equations based on weighted least-squares, 1/2018--12/2020.
• Ling, Leevan, GRF/12304114 [32-14-341] $614,810, Error-optimal methods for PDEs arise from meshless theories, 01/2015-12/2017.
• Liu, Hongyu, Mathematical and computational studies of geomagnetic anomaly detections, HKBU12302919, HK$502,444, 01 January 2020 - 31 December 2022
• Liu, Hongyu, Mathematical analysis on scattering from corner singularities, inverse shape problems and geometric structures of transmission eigenfunctions, HKBU12301218, HK$304,301, 01 September 2018 - 03 March 2020
• Ng, Michael K. P., Preconditioning Methods for Inverse Source Problem in Fractional Diffusion Equations, HKBU12300519, HK$332,261, 01 January 2020 - 31 December 2022
• Ng, Michael K. P., Orthogonal Non-negative Matrix and Tensor Factorizations: Analysis, Algorithms and Applications, HKBU12300218, 172824, HK$456,452, 01 January 2019 - 11 September 2019
• Ng, Michael K. P., Nonlinear Toeplitz-like Matrix Systems in Image Restoration, HKBU12200317, 172732, HK$472,351, 01 January 2018 - 11 September 2019
• Ng, Michael K. P., Sparse Multilinear PageRank: Formulation, Algorithm and Analysis, HKBU12306616 32-16-066, HK$727,647, 1 January 2017 - 31 December 2019
• Tai, Xue-Cheng, RG(R)- RC/17-18/02-MATH, Research on Scientific Computing and Imaging Science, HKBU, HK$500,000, 1 Aug 2017 (24 months).
• Tai, Xue-Cheng, HKBU12300819, Efficient Splitting And Preconditioning Algorithms For Total Variation And Elastica Energy Minimization For Variational Image Processing And Computer Vision, HK$502,444, 01 January 2020 - 31 December 2022
• Tai, Xue-Cheng, Mathematical modelling and analysis of deep neural networks for solving structured differential and integral models, NSFC/RGC Joint Research Scheme, HK$1,177,667, 1 Jan 2020 (36 months).
• Zhu, Lixing, HKBU12302720, Estimation of conditional M-quantile treatment effect, HKD 529,289, 1/1/2021-31/12/2023
• Zhu, Lixing, HKBU12303419, Order determination for large dimensional matrices and its application, HKD 502,444, 1/1/2020-31/12/2022
• Zhu, Lixing, HKBU123028/18, Model checking for regressions: local smoothing-based tests with global smoothing features, HKD 651,273, 1/1/2019-31/12/2021
• Zhu, Lixing, HKBU123017/17, Thresholding double ridge ratio criteria for order determination and their applications, HKD 472,351, 1/1/2018-31/12/2020
• Zhu, Lixing, HKBU123289/16P, Nonparametric generalized likelihood ratio and significant testing for regressions: dimension reduction approaches, HK$ 326,811, 1/1/2017-31/12/2019
• Zhu, Lixing, HKBU123031/15P, Sufficient dimension reduction with mixture multivariate skew-elliptical distributions, HK$ 451,255, 1/1/2016-30/06/2019
• Chiu, Sung Nok, HKBU12317616, Analysis of multivariate spatial point patterns, January 2017 - December 2019, HK$410,371.
• Chiu, Sung Nok, HKBU12301215, Asymptotic methods for spatial point processes, January 2016 - December 2018, HK$451,255.
• Fan, Jun, Mathematical analysis of kernel based modal regression schemes. Early Career Scheme (ECS), Hong Kong, HKD 446,678, September 2018 - August 2021.
• Fan, Jun, Learning theory of Kaczmarz Methods for Phase Retrieval. National Natural Science Foundation of China (NSFC), China, RMB 220,000, January 2019 December 2021.
• Fan, Jun, Weighted tubal decompositions for tensors with applications in image processing. GRF HKBU12301619, HKD 502,444, September 2019 - August 2022.
• Fan, Jun, Analysis of kernel-based learning and testing schemes for large-scale data. GRF HKBU12302819, HKD 332,261, January 2020 - December 2022.
• Fan, Jun, Learning theory of outcome weighted schemes and related topics. GRF HKBU12303220 , HKD 599,861, January 2021 - December 2023.
• Hon, Sean Y. S., ECS 22300921, $951,823, Preconditioning for Toeplitz-like Systems in Evolutionary Differential Equations, 2021-2024
• Hon, Sean Y. S., Startup Allowance for Croucher Study Awards, $500,000, A Fast Iterative Method for Ill-conditioned Toeplitz Systems, 2021-2024
• Kwok, Felix, Domain decomposition based nonlinear preconditioning for optimal control problems, HKBU12300720, HK$599,861
• Kwok, Felix, Adaptive Pipelining Methods for Parallel-in-Time Integration, HKBU12301018, 172828, HK$456,452, 01 January 2019 - 31 December 2021
• Kwok, Felix, Space-time B-methods for the integration of nonlinear parabolic PDEs with blow-up, HKBU12301817, 172744, HK$472,351, 01 January 2018 - 31 December 2020
• Lam, Andrew K. F., Modelling and analysis of diffuse interface models for two-phase micropolar fluid flows. GRF 14303420, HKD 555,754, Jan 2021 to Dec 2023
• Lam, Andrew K. F., On Cahn–Hilliard models with singular potentials and source terms. GRF 14302319, HKD 502,444, Sep 2019 to Aug 2022
• Lam, Andrew K. F., Mathematical studies of a phase field approach to shape optimization. GRF 14302218, HKD 456,452, Jan 2019 to Dec 2021
• Peng, Heng, HKBU 12302615, Revisiting the Generalized Likelihood Ratio Test and Its Related Non-parametric Regression Inference Problems, 1/2016-12/2018, HKD 451,255
• Peng, Heng, HKBU 12303618, Trade-offs between Statistical Efficiency and Complexity of Computation by Discretization and Partial consistency, 1/2019-12/2021, HKD 456,452
• Tong, Tiejun, HKBU12303918, Some new developments in hypothesis testing for high-dimensional data, HK$ 456,452, December 2019-May 2022.
• Tong, Tiejun, Difference-based methods in statistics and their wide applications. National Natural Science Foundation of China, CNY 480,000, January 2017 - December 2020.
• Tong, Tiejun, Data transformation for meta-analysis with application to evidence-based medicine. Health and Medica Add to l Research Fund, Hong Kong, HK$ 376,960, July 2017 - June 2019.