本文针对二维亥姆霍兹核的$N$体问题，提出了一种方向性多尺度算法，并应用于高频散射。该算法遵循[Engquist and Ying, SIAM Journal on Scientific Computing, 29 (4), 2007]中研究三维情况的方法。主要观察到，对于遵循方向抛物几何配置的两个区域，这些区域中的点通过亥姆霍兹核的相互作用近似低秩。我们提出了一种改进的随机化过程来生成低秩表示。基于这些表示，我们以多方向和多尺度的方式组织远场相互作用的计算，以实现最大效率。该算法精确，对于二维散射应用中的问题具有最优的$O(N\log N)$复杂度。我们给出了几个测试例子的数值结果，以说明该算法及其在二维高频散射问题中的应用。
英文摘要：
This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal on Scientific Computing, 29 (4), 2007] where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric configuration, the interaction between the points in these two regions through the Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank representations. Based on these representations, we organize the computation of the far field interaction in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(N\log N)$ complexity for problems from two dimensional scattering applications. We present numerical results for several test examples to illustrate the algorithm and its application to two dimensional high frequency scattering problems.
We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered directional multi-level approximation of the Helmholtz kernel can be applied even on high-frequency levels efficiently. We provide a detailed analysis of the almost linear asymptotic complexity of the presented method. Our numerical experiments are in good agreement with the provided theory.
Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited i…
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）
Perfectly matched layers (PMLs) are formulated and applied to numerically solve nonlocal Helmholtz equations in one and two dimensions. In one dimension, we present the PML modifications for the nonlocal Helmholtz equation with general kernels and theoretically show its effectiveness in some sense. In two dimensions, we give the PML modifications in both Cartesian coordinates and polar coordinates. Based on the PML modifications, nonlocal Helmholtz equations are truncated in one and two dimensional spaces, and asymptotic compatibility schemes are introduced to discretize the resulting truncate…
This paper introduces a parallel directional fast multipole method (FMM) for solving N-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive low-rank criterion than that of the low-frequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between p processes, the overall runtime is essentially O(N log N/p+ p). By the structure …
Boundary element methods for the Helmholtz equation lead to large dense matrices that can only be handled if efficient compression techniques are used. Directional compression techniques can reach good compression rates even for high-frequency problems. Currently there are two approaches to directional compression: analytic methods approximate the kernel function, while algebraic methods approximate submatrices. Analytic methods are quite fast and proven to be robust, while algebraic methods yield significantly better compression rates. We present a hybrid method that combines the speed and re…
We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Pa…
Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media
Hunan Normal University（湖南师范大学）；Southern Methodist University（南方卫理公会大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Indiana University（印第安纳大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
Fast multipole methods (FMM) were originally developed for accelerating $N$-body problems for particle-based methods. FMM is more than an $N$-body solver, however. Recent efforts to view the FMM as an elliptic Partial Differential Equation (PDE) solver have opened the possibility to use it as a preconditioner for a broader range of applications. FMM can solve Helmholtz problems with optimal $\mathcal{O}(N \log N)$ complexity, has compute-bound inner kernels, and highly asynchronous communication patterns. The combination of these features makes FMM an interesting candidate as a preconditioner …
Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions
We present a fast iterative solver for scattering problems in 2D, where a penetrable object with compact support is considered. By representing the scattered field as a volume potential in terms of the Green's function, we arrive at the Lippmann-Schwinger equation in integral form, which is then discretized using an appropriate quadrature technique. The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel. And th…
This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, do…
Kernel methods are a highly effective and widely used collection of modern machine learning algorithms. A fundamental limitation of virtually all such methods are computations involving the kernel matrix that naively scale quadratically (e.g., constructing the kernel matrix and matrix-vector multiplication) or cubically (solving linear systems) with the size of the data set $N.$ We propose the Fast Kernel Transform (FKT), a general algorithm to compute matrix-vector multiplications (MVMs) for datasets in moderate dimensions with quasilinear complexity. Typically, analytically grounded fast mul…
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Department of Computer Science, University of Maryland, College Park（马里兰大学学院公园分校计算机科学系）；Fantalgo, LLC（Fantalgo有限责任公司）；Institute for Advanced Computer Studies, University of Maryland, College Park（马里兰大学学院公园分校高级计算机研究所）
We consider the numerical solution of large-scale M-matrix algebraic Riccati equations with low-rank structures. We derive a new doubling iteration, decoupling the four original iteration formulae in the alternating-directional doubling algorithm. We prove that the kernels in the decoupled algorithm are small M-matrices. Illumined by the highly accurate algorithm proposed by Xue and Li in 2017, we construct the triplet representations for the small M-matrix kernels in a highly accurate doubling algorithm. Illustrative numerical examples will be presented on the efficiency of our algorithm.
