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A fast direct solver for nonlocal operators in wavelet coordinates
arXiv - CS - Numerical Analysis Pub Date : 2020-07-03 , DOI: arxiv-2007.01541 Helmut Harbrecht and Michael Multerer
arXiv - CS - Numerical Analysis Pub Date : 2020-07-03 , DOI: arxiv-2007.01541 Helmut Harbrecht and Michael Multerer
In this article, we consider fast direct solvers for nonlocal operators. The
pivotal idea is to combine a wavelet representation of the system matrix,
yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The
latter drastically reduces the fill-in during the factorization of the system
matrix by means of a Cholesky decomposition or an LU decomposition,
respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in
the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments
for different highly relevant applications of nonlocal operators: We consider
(i) the direct solution of boundary integral equations in three spatial
dimensions, issuing from the polarizable continuum model, (ii) a parabolic
problem for the fractional Laplacian in integral form and (iii) the fast
simulation of Gaussian random fields.
中文翻译:
小波坐标中非局部算子的快速直接求解器
在本文中,我们考虑非局部运算符的快速直接求解器。关键思想是将系统矩阵的小波表示与嵌套解剖排序方案相结合,产生一个准稀疏矩阵。后者分别通过 Cholesky 分解或 LU 分解大大减少了系统矩阵分解过程中的填充。这样,我们最终得到了压缩系统矩阵的精确逆,矩阵中的非零条目的数量仅适度增加。为了说明该方法的有效性,我们对非局部算子的不同高度相关应用进行了数值实验:我们考虑 (i) 三个空间维度中边界积分方程的直接解,源自可极化连续模型,
更新日期:2020-07-06
中文翻译:
小波坐标中非局部算子的快速直接求解器
在本文中,我们考虑非局部运算符的快速直接求解器。关键思想是将系统矩阵的小波表示与嵌套解剖排序方案相结合,产生一个准稀疏矩阵。后者分别通过 Cholesky 分解或 LU 分解大大减少了系统矩阵分解过程中的填充。这样,我们最终得到了压缩系统矩阵的精确逆,矩阵中的非零条目的数量仅适度增加。为了说明该方法的有效性,我们对非局部算子的不同高度相关应用进行了数值实验:我们考虑 (i) 三个空间维度中边界积分方程的直接解,源自可极化连续模型,