相关论文: High Accuracy Method for Integral Equations with D…
We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the…
In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme…
Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and…
A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic…
We present SQDFT: a large-scale parallel implementation of the Spectral Quadrature (SQ) method for $\mathcal{O}(N)$ Kohn-Sham Density Functional Theory (DFT) calculations at high temperature. Specifically, we develop an efficient and…
Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior…
The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra…
Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a…
The paper is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such…
In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels,…
Kernel matrices are a key quantity in kernel-based approximation, and important properties such as stability and algorithmic convergence can be analyzed with their help. In this work we refine a multivariate Ingham-type theorem, which is…
We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
Most Fredholm integral equations involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common…
In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of…
In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$…
Solutions of Fredholm integral equations of the second kind with oscillatory kernels likely exhibit oscillation. Standard numerical methods applied to solving equations of this type have poor numerical performance due to the influence of…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
We present a novel spectral method for the Allen-Cahn equation on spheres, eliminating the reliance on conventional quadrature exactness conditions. By replacing these conditions with a restricted isometry relation derived from…
Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values…