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Multipolynomial Monte Carlo Trace Estimation

Paul Lashomb, Ronald B. Morgan, Travis Whyte, Walter Wilcox

Abstract

In lattice QCD the calculation of disconnected quark loops from the trace of the inverse quark matrix has large noise variance. A multilevel Monte Carlo method is proposed for this problem that uses different degree polynomials on a multilevel system. The polynomials are developed from the GMRES algorithm for solving linear equations. To reduce orthogonalization expense, the highest degree polynomial is a composite or double polynomial found with a polynomial preconditioned GMRES iteration. Matrix deflation is used in three different ways: in the Monte Carlo levels, in the main solves, and in the deflation of the highest level double polynomial. A numerical comparison with optimized Hutchinson is performed on a quenched \(24^4\) lattice. The results demonstrate that the new Multipolynomial Monte Carlo method can significantly improve the trace computation for matrices that have a difficult spectrum due to small eigenvalues.}

Multipolynomial Monte Carlo Trace Estimation

Abstract

In lattice QCD the calculation of disconnected quark loops from the trace of the inverse quark matrix has large noise variance. A multilevel Monte Carlo method is proposed for this problem that uses different degree polynomials on a multilevel system. The polynomials are developed from the GMRES algorithm for solving linear equations. To reduce orthogonalization expense, the highest degree polynomial is a composite or double polynomial found with a polynomial preconditioned GMRES iteration. Matrix deflation is used in three different ways: in the Monte Carlo levels, in the main solves, and in the deflation of the highest level double polynomial. A numerical comparison with optimized Hutchinson is performed on a quenched lattice. The results demonstrate that the new Multipolynomial Monte Carlo method can significantly improve the trace computation for matrices that have a difficult spectrum due to small eigenvalues.}
Paper Structure (10 sections, 13 equations, 1 figure, 1 table)

This paper contains 10 sections, 13 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Subtraction Methods
  3. The GMRES Polynomial
  4. Relative Variance
  5. Reducing Costs Through Double Polynomials
  6. Multipolynomial Monte Carlo
  7. Deflation for the Monte Carlo
  8. Algorithm and Testing
  9. Conclusions
  10. Acknowledgements

Figures (1)

  • Figure 1: Log-averaged standard relative variance of the scalar operator using polynomial subtraction against the degree of the subtraction polynomial. Lattice volumes $4^3 \times 4$, $8^3 \times 8$, $12^3 \times 32$, and $24^3 \times 32$ are shown, each averaged over 10 quenched configurations at $\beta = 6.0$ and $\kappa = 0.1570$.