Fast, Adaptive, High-Order Accurate Discretization of the Lippmann-Schwinger Equation in Two Dimensions
Fast, Adaptive, High-Order Accurate Discretization of the Lippmann-Schwinger Equation in Two Dimensions
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二维 Lippmann-Schwinger 方程的快速、自适应、高阶精确离散化
DOI：
10.1137/15m102455x
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
L. Greengard
中科院分区：
文献类型：
--
作者：
Sivaram Ambikasaran;C. Borges;Lise;L. Greengard
提出了一种二维散射问题的快速直接求解器，其中入射波入射到具有紧密支承的可穿透介质上。我们用体势来表示散射场，体势的核是外部区域的出射格林函数。将这种表示嵌入到控制偏微分方程组中，我们得到了Lippmann-Schwinger型积分方程。这里的主要贡献是开发了一种基于四叉树数据结构的自动自适应高阶精确离散化，它提供了对离散化系统矩阵的任意元素的快速访问。这允许直接应用最先进的算法来构造解操作符的压缩版本。这些求解器通常需要$O(N^{3/2})$work，其中$N$表示自由度数。我们演示了该方法对各种问题在低频和高频区域的性能。
We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann--Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad-tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both low and high frequency regimes.