In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.
In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points (interior, exterior, or exactly on the boundary) inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. Compared to the original Quadrature by Expansion (QBX) method, the QB2X method explicitly includes the n…
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansion which are then evaluated to compute the near or on-surface value of the integral. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon t…
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both d…
We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX (`GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion (`QBX') using local expansions formed via the Fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from $O((p+1)^{2})$ to $O(p+1)$ in three dimensions, without …
Numerical simulations with rigid particles, drops or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper we present quadratu…
We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provide valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundar…
A Range-Deflated Stochastic Lanczos Quadrature Method for Large-Scale Log-Determinant Estimation
The University of Dodoma（多多马大学）；African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Western Norway University of Applied Sciences（挪威西部应用科学大学）；University of Cape Town（开普敦大学）
An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, a…
To evaluate the Hadamard finite-part integrals accurately, a novel interpolatory-type quadrature is proposed in this article. In our approach, numerical divided difference is utilized to represent the high order derivatives of the integrated function, which make it possible to reduced the numerical quadrature into a concise formula based on the cycle index for symmetric group. In addition, convergence analysis is presented and the error estimation is given. Numerical results are presented on cases with different weight functions, which substantiate the performance of the proposed method.
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…
We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012]…
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…
While fast multipole methods (FMMs) are in widespread use for the rapid evaluation of potential fields governed by the Laplace, Helmholtz, Maxwell or Stokes equations, their coupling to high-order quadratures for evaluating layer potentials is still an area of active research. In three dimensions, a number of issues need to be addressed, including the specification of the surface as the union of high-order patches, the incorporation of accurate quadrature rules for integrating singular or weakly singular Green's functions on such patches, and their coupling to the oct-tree data structures on w…
Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This paper presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centrepiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion (QBX), accelerated using a precomputation scheme. The…
Approximate solutions of Urysohn integral equations using projection methods involve integrals which need to be evaluated using a numerical quadrature formula. It gives rise to the discrete versions of the projection methods. For $r \geq 1,$ a space of piece-wise polynomials of degree $\leq r - 1$ with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at $r$ Gauss points. Asymptotic expansion for the iterated modified projection solution is available in literature. In this paper, we obtain an asymptotic expan…
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学圣克利门特奥赫里斯基数学与信息学院）；Faculty of Pharmacy, Medical University of Sofia（索非亚医科大学药学院）
Estimation of errors of quadrature formula for singular integrals of Cauchy type with special forms
Institute of Mathematics, Academy of Science of Uzbekistan（乌兹别克斯坦科学院数学研究所）；Department of Mathematics, Faculty of Science, Universiti Putra Malaysia（马来西亚博特拉大学理学院数学系）
A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathema…
Dispersion optimized quadratures for isogeometric analysis
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷计算研究所）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织矿产资源部）
In this paper, we consider a zero-order stochastic oracle model of estimating definite integrals. In this model, integral estimation methods may query an oracle function for a fixed number of noisy values of the integrand function and use these values to produce an estimate of the integral. We first show that the information-theoretic error lower bound for estimating the integral of a $d$-dimensional function over a region with $l_\infty$ radius $r$ using at most $T$ queries to the oracle function is $Ω(2^d r^{d+1}\sqrt{d/T})$. Additionally, we find that the Gaussian Quadrature method under th…
We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nodes close to the singularity are judiciously corrected based on these expansions. Then we define the composite corrected trapezoidal rules for a larger family of functions using series expansions around the point singularity and applying corrected tra…
Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs
National Center for Applied Mathematics, Tianjin University, China（应用数学国家中心，天津大学，中国）；School of Mechanical and Aerospace Engineering, Jilin University, China（机械与 aerospace 工程学院，吉林大学，中国）
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…
This paper deals with the estimation of the quadrature error of a Gaussian formula for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. For this purpose, in this work the averaged and generalized averaged Gaussian rules are employed, together with a tentative a priori approximation of the error. The numerical examples confirm the reliability of these approaches.
