High-degree polynomial noise subtraction and Multipolynomial Monte Carlo trace estimation in Lattice QCD.

In lattice Quantum Chromodynamics, the determination of physical quantities containing disconnected loop effects is very computationally expensive. For this reason, the trace of the inverted Wilson-Dirac matrix required for such measurements are performed stochastically through Monte Carlo methods. These trace estimators, however, are generally flooded with noise and require the use of noise subtraction methods to improve the signal-to-noise ratio. In this work, we examine the variance reduction effects of using high-degree GMRES polynomials. We work in the quenched approximation near κcrit for multiple lattice volumes as a demonstration of the methods. We observe a significant reduction in the variance of the scalar operator for such high-degree polynomials.

Performing the full trace estimation using these high-degree subtraction polynomials, however, requires computing the trace of these same high-degree polynomials. To make the noise subtracted trace estimator more cost effective, we additionally outline our new Multipolynomial Monte Carlo method and demonstrate several ways that it can be made more efficient. Some of these improvements include the use of double polynomials, polynomial preconditioning, and several novel applications of deflation. In order to demonstrate the performance of our Multipolynomial Monte Carlo method, we have simultaneously developed a very efficient form of Hutchinson’s trace estimator and show that our multi-polynomial approach successfully outperforms it.