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：

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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，