Hierarchical probing for estimating the trace of the matrix inverse on toroidal lattices

The standard approach for computing the trace of the inverse of a very large, sparse matrix $A$ is to view the trace as the mean value of matrix quadratures, and use the Monte Carlo algorithm to estimate it. This approach is heavily used in our motivating application of Lattice QCD. Often, the elements of $A^{-1}$ display certain decay properties away from the non zero structure of $A$, but random vectors cannot exploit this induced structure of $A^{-1}$. Probing is a technique that, given a sparsity pattern of $A$, discovers elements of $A$ through matrix-vector multiplications with specially designed vectors. In the case of $A^{-1}$, the pattern is obtained by distance-$k$ coloring of the graph of $A$. For sufficiently large $k$, the method produces accurate trace estimates but the cost of producing the colorings becomes prohibitively expensive. More importantly, it is difficult to search for an optimal $k$ value, since none of the work for prior choices of $k$ can be reused.