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In this paper, we present a fast and accurate numerical scheme for the solution of fifth-order boundary-value problems. We apply the reproducing kernel Hilbert space method (RKHSM) for solving this problem. The analytic results of the…
This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical…
Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point…
I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the…
Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a…
In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that…
There have been increasing interests on the Volterra series identification with the kernel-based regularization method. The major difficulties are on the kernel design and efficiency of the corresponding implementation. In this paper, we…
A new Hardy space Hardy space approach of Dirichlet type problem based on Tikhonov regularization and Reproducing Hilbert kernel space is discussed in this paper, which turns out to be a typical extremal problem located on the upper…
We present a large family of Spin(p,q)-valued discrete spectral problems. The associated discrete nets generated by the so called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These…
The authors propose a Nystrom method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability…
In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced…
Ab initio methods aim to solve the nuclear many-body problem with controlled approximations. Virtually exact numerical solutions for realistic interactions can only be obtained for certain special cases such as few-nucleon systems. Here we…
This paper presents a re-formulation of the boundary integral method (BIM) for the Debye-Huckel model of molecular and colloidal electrostatics that removes the mathematical singularities that have been accepted as an intrinsic part of the…
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…
A major problem of kernel-based methods (e.g., least squares support vector machines, LS-SVMs) for solving linear/nonlinear ordinary differential equations (ODEs) is the prohibitive $O(an^3)$ ($a=1$ for linear ODEs and 27 for nonlinear…
Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth…
We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of $H^1$ into the image and the kernel of some novel…
We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source…