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.

在本文中，我们考虑用于非本地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形式（产生准稀疏矩阵）与嵌套解剖排序方案相结合。后者分别通过Cholesky分解或LU分解大大减少了系统矩阵分解期间的填充。这样，我们最终得到压缩系统矩阵的确切逆，而矩阵中非零项的数量仅适度增加。

为了说明该方法的有效性，我们针对非局部算子的不同高度相关的应用进行了数值实验：我们考虑（i）在三个空间维度上边界积分方程的直接解，这是从可极化连续体模型产生的，（ii）抛物线分数形式的拉普拉斯算子问题和（iii）高斯随机场的快速仿真。