Probing is a general technique that is used to reduce the variance of the Hutchinson stochastic estimator for the trace of the inverse of a large, sparse matrix $A$. The variance of the estimator is the sum of the squares of the off-diagonal elements of $A^{-1}$. Therefore, this technique computes probing vectors that when used in the estimator they annihilate the largest off-diagonal elements. For matrices that display decay of the magnitude of $|A^{-1}_{ij}|$ with the graph distance between nodes $i$ and $j$, this is achieved through graph coloring of increasing powers $A^p$. Equivalently, when a matrix stems from a lattice discretization, it is computationally beneficial to find a distance-$p$ coloring of the lattice. In this work, we study probing for the more general problem of computing the trace of a permutation of $A^{-1}$, say $PA^{-1}$, motivated from Lattice QCD where we need to construct "disconnected diagrams" to extract flavor-separated Generalized Parton functions. In Lattice QCD, where the matrix has a 4D toroidal lattice structure, these non-local operators correspond to a $PA^{-1}$ where $P$ is the permutation relating to some displacement in one or more dimensions. We focus on a single dimension displacement ($k$) but our methods are general. We show that probing on $A^p$ or $(PA)^p$ do not annihilate the largest magnitude elements. To resolve this issue, our displacement-based probing works on $PA^p$ using a new coloring scheme that works directly on appropriately displaced neighborhoods on the lattice. We prove lower bounds on the number of colors needed, and study the effect of this scheme on variance reduction, both theoretically and experimentally on a real-world Lattice QCD calculation. We achieve orders of magnitude speedup over the un-probed or the naively probed methods.

翻译：检验是一种普通技术, 用来减少 Hutchinson 的离位值差异。 对于显示 $A 和 $ 美元 之间正方位距离的 Exchantistial 估测器, 以追踪 $A 。 估测器的差异是 $A =-1 $ 美元 的离位元素的平方数之和 。 因此, 这个技术计算了在测量器中用于清除最大的离位元素的矢量。 对于显示 $A =-1 美元 和 $ $ 美元 之间正方位距离下降的基数矩阵, 这一点通过 美元 美元 美元 美元 美元 和 美元 美元 美元 的正方位偏差的图形颜色来实现。 当一个离位的离位模型来自 $A = 1 = 1 美元 和 美元 美元 的离位方位值计算器, 也就是我们用 美元 = 美元 美元 美元 基质 的算算算法 。