HKBU-SJTU Joint Workshop
on Scientific Computing

This talk begins with an introduction of kernel methods and will cover some popular numerical methods for solving PDEs including asymmetric collocation and meshfree finite difference methods.

A matrix A is monotonically positive (MP) if there exists a matrix U such that A =UU^T and each column of U is monotonically nonincreasing or nondecreasing. We propose a semidefinite algorithm for checking whether or not a matrix is MP. If it is not MP, a certificate for it can be obtained; if it is MP, an MP-decomposition can be obtained. Some computational experiments are presented to show how to do this.

In this talk, we will discuss and address the computational issues in many interior point algorithms. We will start by addressing some convergent interior point methods and continuous trajectories. Then some numerical difficulties and challenges resulting from these methods and trajectories will be raised and discussed. Some preliminary numerical results on certain numerical algorithms will be also reported.

The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this talk, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH outperforms existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius model.

A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The discretization in time is based on the $P_1$-continuous discontinuous Galerkin (CDG) method, while the spatial discretization on the Crouziex-Raviart (CR) element. It is proved that the error of the displacement (resp. velocity) in the energy norm (resp. $L^2$ norm) is bounded by $O(h+k)$ (resp. $O(h^2+k)$), where $h$ and $k$ denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lam\'{e} coefficients of the elastic material under discussion. Several numerical results are reported to illustrate numerical performance of the proposed method.

We develop a modified Poisson-Nernst-Planck model to include Coulomb many-body properties in electrolytes, which also takes the ion-size effect into account and is expected to provide more accurate prediction for ion dynamics with microscopic confinement. Asymptotic expansions are performed to remove the multiscale properties of the equations and also used to understand dielectric properties near interfaces. Furthermore, we discuss numerical strategies to solve the resulted PDEs and show numerical results to demonstrate the performance of our numerical methods.

In this talk, I will present two versions (one has second-order accuracy and another one has fourth-order accuracy) of the kernel-free boundary integral method for the biharmonic equation on complex domains. This method is a generalization of the traditional boundary integral method. It does not need to know analytical expressions of the kernel or associated Green's function of the differential operator and evaluate boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids with FFT-based fast solvers. The method has several advantages over the traditional finite element based biharmonic solvers that work with a saddle-point formulation and unstructured grids. Numerical examples will be included to demonstrate accuracy and efficiency of the method.

In this talk I will present our recent finding on the intrinsic geometric structure of interior transmission eigenfunctions arising in wave scattering theory. We numerically show that the aforementioned geometric structure can be very delicate and intriguing. The major findings can be roughly summarized as follows. If there is a cusp, i.e. a discontinuity of the surface tangent on the support of the underlying potential function, then the interior transmission eigenfunction vanishes near the cusp if its interior angle is less than Pi, whereas the interior transmission eigenfunction localizes near the cusp if its interior angle is bigger than Pi. Furthermore, we show that the vanishing and blowup orders are inversely proportional to the interior angle of the cusp: the sharper the corner, the higher the convergence order.

I will talk about some recent progress on multiscale coupling methods for materials defects. Based on an unified analytical framework, we have studied a priori and a posteriori error estimate for atomisitc/continuum (a/c) coupling, higher order continuum model, and furthermore, extensions to electronic models and soft materials such as nematic liquid crystal.

Functional data analysis is concerned with inherently infinite dimensional data such as curves or images. It attracts more and more attentions because of its successful applications in many areas such as neuroscience and econometrics. We consider a class of regularization methods called spectral algorithms for functional linear regression within the framework of reproducing kernel Hilbert space. The proposed estimators can achieve the minimax optimal rates of convergence. Despite of the infinite dimensional nature of functional data, we show that the algorithms are easily implementable.

In this talk, we consider a class of domain decomposition methods, known as waveform relaxation (WR) methods, for solving time-dependent PDEs numerically on many different processors in parallel. WR methods are distinctive in that a typical subdomain problem is posed in both space and time; each iteration requires the parallel solution of these space-time subproblems, followed by an exchange of interface data defined over the whole time window. An often cited advantage of WR methods is that they allow each subdomain to use a different spatial and temporal grid that is adapted to the dynamics of the local subproblem.

In this talk, I will first present some new results on the convergence of WR methods of the Neumann-Neumann type. Next, I will discuss two ways of introducing parallelism in time to the basic WR method. The first approach uses a fixed time-window size and yields an algorithm that is mathematically equivalent to the original WR method. The second one, on the other hand, chooses time-window size dynamically based on how many free processors are available; this leads to a method with improved convergence behaviour. We demonstrate the effectiveness of both approaches by comparing their running times against those obtained from classical time-stepping methods, where the same number of processors is used to parallelize in space only.

In this talk, we shall consider a class of inverse problems of simultaneously recovering an embedded source and it surrounding mediums. This type of inverse problems arises from a variety of applications including thermoacoustic and photoacoustic tomography, brain imaging and geomagnetic detection technology. I shall talk about the recent progress of our study on those inverse problems.