Related papers: High Accuracy Method for Integral Equations with D…
This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend…
This paper is concerned with the direct and inverse acoustic or electromagnetic scattering problems by a locally perturbed, perfectly reflecting, infinite plane (which is called a locally rough surface in this paper). We propose a novel…
We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural…
The boundary integral method is an efficient approach for solving time-harmonic acoustic obstacle scattering problems. The main computational task is the evaluation of an oscillatory boundary integral at each discretization point of the…
A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when…
We present a real-space formulation for coarse-graining Kohn-Sham Density Functional Theory that significantly speeds up the analysis of material defects without appreciable loss of accuracy. The approximation scheme consists of two steps.…
We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete…
Based on the superconvergent approximation at some point (depending on the fractional order $\alpha$, but not belonging to the mesh points) for Gr\"{u}nwald discretization to fractional derivative, we develop a series of high order…
We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three dimensional space. Our approach relies on new integral formulations as well as associated high-order quadrature rules.…
Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the…
We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex…
The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…
The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density,…
A new computational algorithm, the discrete singular convolution (DSC), is introduced for computational electromagnetics. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied.…
We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the…
The nonlocal Cahn-Hilliard equation provides a natural extension of the classical model for phase separation by incorporating long-range interactions through a singular convolution kernel. While this formulation admits a rich existence and…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
This paper present a numerical method for solving nonlinear Fredholm integral equations. The method is based upon Newton type approximations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
An infinitely wide model is a weighted integration $\int \varphi(x,v) d \mu(v)$ of feature maps. This model excels at handling an infinite number of features, and thus it has been adopted to the theoretical study of deep learning. Kernel…
We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the…