Related papers: Quadrature error estimates for layer potentials ev…
In this paper, we consider the Gauss quadrature formulae corresponding to some modifications of anyone of the four Chebyshev weights, considered by Gautschi and Li in \cite{gauli}. As it is well known, in the case of analytic integrands,…
The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial…
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 detection of insufficiently resolved or ill-conditioned integrand structures is critical for the reliability assessment of the quadrature rule outputs. We discuss a method of analysis of the profile of the integrand at the quadrature…
Accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations. Quadrature by expansion (QBX) is a recently introduced method that can offer high accuracy for…
Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and…
The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform…
We introduce a quadrature scheme--QBKIX--for the high-order accurate evaluation of layer potentials associated with general elliptic PDEs near to and on the domain boundary. Relying solely on point evaluations of the underlying kernel, our…
Recently, Trefethen (SIAM Review 50 (2008), 67--87) and Xiang and Bornemann (SIAM J. Numer. Anal. 50 (2012), 2581--2587) investigated error bounds for n-point Gauss and Clenshaw-Curtis quadrature for the Legendre weight with integrands…
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…
Overtaking in high-speed autonomous racing demands precise, real-time estimation of collision risk; particularly in wheel-to-wheel scenarios where safety margins are minimal. Existing methods for collision risk estimation either rely on…
We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a…
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…
Fourier feature approximations have been successfully applied in the literature for scalable Gaussian Process (GP) regression. In particular, Quadrature Fourier Features (QFF) derived from Gaussian quadrature rules have gained popularity in…
We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete {\em a priori} error analysis for the case of…
Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of…
We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of…
This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a…
3D Gaussian Splatting has garnered extensive attention and application in real-time neural rendering. Concurrently, concerns have been raised about the limitations of this technology in aspects such as point cloud storage, performance, and…
Gauss--Christoffel quadrature is a fundamental method for numerical integration, and its convergence analysis is closely related to the decay of Chebyshev expansion coefficients. Classical estimates, including those due to Trefethen, are…