Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids) or when the solution is needed in the bulk, nearly-singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case where each object is "simple", ie, its smooth boundary needs only moderately many nodes. We present a kernel-independent method needing only an upsampled smooth surface quadrature, and one dense factorization, for each distinct shape. No (near-)singular quadrature rules are needed. The resulting effective sources are drop-in compatible with fast algorithms, with no local corrections nor bookkeeping. Our extensive numerical tests include 2D FMM-based Helmholtz and Stokes BVPs with up to 1000 objects (281000 unknowns), and a 3D Laplace BVP with 10 ellipsoids separated by $1/30$ of a diameter. We include a rigorous analysis for analytic data in 2D and 3D.
We propose a method for generating nodes for kernel quadrature by a point-wise gradient descent method. For kernel quadrature, most methods for generating nodes are based on the worst case error of a quadrature formula in a reproducing kernel Hilbert space corresponding to the kernel. In typical ones among those methods, a new node is chosen among a candidate set of points in each step by an optimization problem with respect to a new node. Although such sequential methods are appropriate for adaptive quadrature, it is difficult to apply standard routines for mathematical optimization to the pr…
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 potentials belonging to the modified Helmholtz, biharmonic and Stokes equations. These equations depend on a parameter, denoted $α$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The r…
Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many …
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 potentials belonging to the modified Helmholtz, modified biharmonic and modified Stokes equations. These equations depend on a parameter, denoted $α$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the sour…
We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.
This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase…
High-order kernel regularization of singular and hypersingular Helmholtz boundary integral operators
POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris（POEMS，CNRS，Inria，ENSTA巴黎，巴黎理工学院）；Department of Applied Mathematics, University of Twente（特温特大学应用数学系）；Department of Mathematics, Farmingdale State College, SUNY（萨福克大学法明代尔州立学院数学系）
This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing…
The kernel herding algorithm is used to construct quadrature rules in a reproducing kernel Hilbert space (RKHS). While the computational efficiency of the algorithm and stability of the output quadrature formulas are advantages of this method, the convergence speed of the integration error for a given number of nodes is slow compared to that of other quadrature methods. In this paper, we propose a modified kernel herding algorithm whose framework was introduced in a previous study and aim to obtain sparser solutions while preserving the advantages of standard kernel herding. In the proposed al…
University of Warwick, Department of Statistics（沃里克大学统计系）；Imperial College London, Department of Mathematics（伦敦帝国理工学院数学系）；Newcastle University, School of Mathematics（新castle大学数学系）；University of Technology Sydney, School of Mathematical（悉尼技术大学数学与物理科学学院）；The Alan Turing Institute for Data Science（数据科学阿尔图里研究所）
The well-known Caputo fractional derivative and the corresponding Caputo fractional integral occur naturally in many equations that model physical phenomena under inhomogeneous media. The relationship between the two fractional terms can be readily obtained by applying the Laplace transform to a given equation. We seek to numerically approximate Caputo fractional integrals using a Taylor series expansion for convolution integrals. This naturally extends into being able to approximate convolution integrals for a wider class of convolution integral kernels $K(t-s)$. One of the main advantages un…
This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger equation and to a classical system on a cylindrical lattice.
We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order $N$ calculation for $N$ time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vallée-Poussin sums for a semi-analytical exponential expansion of a general kernel, and a model reduction technique for the minimization of the number of exponentials under given error tolerance. We employ the SOE expansion for the finite pa…
A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -- for choosing node locations -- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove al…
This paper describes a trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured Trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulae that involve the Riemann zeta function in 2D and the Epstein zeta functions in 3D. These expansions are exp…
Monte Carlo methods on compact complex manifolds using Bergman kernels
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg（斯特拉斯堡大学）；Université de Lille, CNRS, Centrale Lille UMR 9189 – CRIStAL（里尔大学）
Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions
Center for Computational Mathematics, Flatiron Institute, Simons Foundation（计算数学中心，Flatiron研究所，西蒙斯基金会）；Institute for Theoretical Sciences, Westlake University（理论科学研究所，西湖大学）
Numerical solution of Volterra integral equations of the first kind with discontinuous kernels
Irkutsk National Research Technical University（伊尔库茨克国立技术研究大学）；Penza State University（奔萨国立大学）；Energy Systems Institute of Russian Academy of Sciences（俄罗斯科学院能源系统研究所）
Quadrature-Enhanced Monte Carlo fPINN Method for High-Dimensional Fractional PDEs
Central China Normal University（华中师范大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
