Hierarchical Probing for Estimating the Trace of the Matrix Inverse on Toroidal Lattices

Abstract

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.First, we introduce the idea of hierarchical probing that produces distance-2 i colorings for a sequence of distances 2 0 , 2 1 , . . . , 2 m , up to the diameter of the graph. To achieve this, we do not color the entire graph, but at each level, i, we compute the distance-1 coloring independently for each of the node-groups associated with a color of the distance-(2 i−1 ) coloring. Second, based on this idea, we develop an algorithm for uniform, toroidal lattices that simply applies bit-arithmetic on local coordinates to produce the hierarchical permutation. Third, we provide an algorithm for choosing an appropriate sequence of Hadamard and Fourier vectors, so that earlier vectors in the sequence correspond to hierarchical probing vectors of smaller distances. This allows us to increase the number of probing vectors until the required accuracy is achieved.Several experiments show that when a decay structure exists in the matrix, our algorithm finds it and approximates the trace incrementally, starting with the most important contributions. We have observed up to an order of magnitude speedup over the standard Monte Carlo.

Variance reduction with practical all-to-all lattice propagators

“…[24]. This so-called "hierarchical probing' elegantly allows to reuse the computations of prior choices of p, and furthermore hints at smaller variances for fixed cost.…”

“…eqs. (24,25); otherwise no signal is observed. Moreover, it can be seen that in both cases the expected rate of convergence with increasing number of configurations taken in the gauge average is observed -i.e.…”

“…Direct comparison with recent works similarly addressing all-to-all techniques (see e.g. [23,24,25,26,27]) are not straightforward.…”

New Physics Searches from Nucleon Matrix Elements in Lattice QCD

“…The disconnected diagram has been neglected until recently because it is very noisy and expensive to compute. During the last few years a number of groups are studying various techniques for its computation [13][14][15][16][17][18] and results at the physical point already appear in the literature [6,19,20] (see Ref. [8] for a recent review).…”

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“…The vertical dotted line indicates the physical pion mass. [20,21], hierarchical probing [68] truncated solver, for loops one-end trick [69] hopping parameter expansion [20,21] R(g For the light-quark disconnected contribution to electromagnetic form factors, a clear nonzero signal was found by LHPC [66]. For the proton G E and G M , these were less than 1% of the connected part (positive for G E , negative for G M ).…”