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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 usedin our motivating application of Lattice QCD. Often, the elements of $A^{-1}$display certain decay properties away from the non zero structure of $A$, butrandom vectors cannot exploit this induced structure of $A^{-1}$. Probing is atechnique that, given a sparsity pattern of $A$, discovers elements of $A$through matrix-vector multiplications with specially designed vectors. In thecase of $A^{-1}$, the pattern is obtained by distance-$k$ coloring of the graphof $A$. For sufficiently large $k$, the method produces accurate traceestimates but the cost of producing the colorings becomes prohibitivelyexpensive. 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.

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