SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1459-A1485, January 2020.
We present extensions to hierarchical probing, a method developed in [A. Stathopoulos, J. Laeuchli, and K. Orginos, SIAM J. Sci. Comput., 35 (2013), pp. S299--S322] to reduce the variance of the Monte Carlo estimation of the trace or the diagonal of the inverse of a large, sparse matrix. In that context, probing is a method to determine the largest-in-magnitude elements of the matrix inverse and then annihilate their contributions to the variance by solving linear systems with appropriate probing vectors. It typically involves coloring the graph of $A^n$, since this matches the sparsity structure of a polynomial approximation to $A^{-1}$. This is equivalent to distance-$n$ coloring of $A$, i.e., determining which nodes are connected to one other at distance $ \leq n$. For matrices that display a Green's function decay, $n$ is small, which reduces the number of linear systems to be solved. Our hierarchical probing method was developed for matrices with a lattice structure, where distance-$n$ coloring and the generation of probing vectors can be performed far more efficiently and in a way so that earlier vectors are subsets of vectors generated later in the process, meaning that it is simple to continue probing if additional accuracy is needed. However, this method worked only on lattices with dimension lengths that were powers of two. In this paper we extend the method to work on lattices of arbitrary dimension lengths, which is theoretically more challenging. Additionally, we expand the idea to a multilevel, hierarchical probing heuristic for matrices with any undirected graph structure that matches the performance of classical probing but with tractable memory requirements.

中文翻译：

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扩展层次探测以计算矩阵逆轨迹

SIAM科学计算杂志，第42卷，第3期，第A1459-A1485页，2020年1月。
我们提出了对分层探测的扩展，这是在[A. Stathopoulos，J。Laeuchli和K.Orginos，SIAM J. Sci。[Comput。35（2013），第S299-S322页]，以减小轨迹的蒙特卡洛估计的方差或大型稀疏矩阵的逆的对角线。在这种情况下，探测是一种确定矩阵逆的幅值最大的元素，然后通过使用适当的探测向量求解线性系统，消除其对方差的贡献的方法。它通常涉及给$ A ^ n $的图着色，因为这与多项式近似的稀疏结构匹配到$ A ^ {-1} $匹配。这等效于$ A $的距离-$ n $着色，即确定哪些节点在距离$ \ leq n $处彼此连接。对于显示格林函数衰减的矩阵，$ n $小，这减少了要解决的线性系统的数量。我们的分层探测方法是针对具有点阵结构的矩阵开发的，其中距离-$ n $着色和探测向量的生成可以更加有效地进行，并且以某种方式使较早的向量成为此过程中稍后生成的向量的子集，这意味着如果需要更高的精度，继续进行探测很简单。但是，此方法仅适用于尺寸长度为2的幂的晶格。在本文中，我们将方法扩展到可处理任意尺寸长度的晶格，这在理论上更具挑战性。此外，我们将这种想法扩展到针对矩阵的多层次，分层探测启发式方法，该矩阵具有与经典探测的性能相匹配但具有可控内存需求的任何无向图结构。