Related papers: Error Analysis of Nodal Meshless Methods
In this paper, we employ a space-time finite element method to discretize the parabolic initial-boundary value problem and extend its error analysis with refined estimates on unstructured space-time meshes. We establish higher-order…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
We develop a universally applicable embedded boundary finite difference method, which results in a symmetric positive definite linear system and does not suffer from small cell stiffness. Our discretization is efficient for the wave, heat…
This study focuses on solving group zero-norm regularized robust loss minimization problems. We propose a proximal Majorization-Minimization (PMM) algorithm to address a class of equivalent Difference-of-Convex (DC) surrogate optimization…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
We are interested in mesh-free formulas based on the Monte-Carlo methodology for the approximation of multi-dimensional integrals, and we investigate their accuracy when the functions belong to a reproducing-kernel space. A kernel typically…
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
We propose improvements in numerical evaluation of symmetric stable density and its partial derivatives with respect to the parameters. They are useful for more reliable evaluation of maximum likelihood estimator and its standard error.…
The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von K\'{a}rm\'{a}n equations.…
In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a…
We consider a singularly perturbed semilinear boundary value problem of a general form that allows various types of turning points. A solution decomposition is derived that separates the potential exponential boundary layer terms. The…
We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter $0< \varepsilon \ll…
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates,…
The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the…
Many applications require minimizing a family of optimization problems indexed by some hyperparameter $\lambda \in \Lambda$ to obtain an entire solution path. Traditional approaches proceed by discretizing $\Lambda$ and solving a series of…
The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local…
We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers…
Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance…