Preconditioning the non-relativistic many-fermion problem

TL;DR

This paper compares three preconditioning strategies for solving linear problems in many-fermion Monte Carlo algorithms, finding that BiCGStab is the most efficient method for the fermion matrix of the unitary Fermi gas.

Contribution

It provides a performance comparison of Chebyshev, strong-coupling, and weak-coupling preconditioners using CG and BiCGStab solvers in the context of many-fermion Monte Carlo methods.

Findings

BiCGStab outperforms other strategies in efficiency.

BiCGStab reduces the number of iterations and matrix-vector operations.

Preconditioning strategy significantly impacts solver performance.

Abstract

Preconditioning is at the core of modern many-fermion Monte Carlo algorithms, such as Hybrid Monte Carlo, where the repeated solution of a linear problem involving an ill-conditioned matrix is needed. We report on a performance comparison of three preconditioning strategies, namely Chebyshev polynomials, strong-coupling approximation and weak-coupling expansion. We use conjugate gradient (CG) on the normal equations as well as stabilized biconjugate gradient (BiCGStab) as solvers and focus on the fermion matrix of the unitary Fermi gas. Our results indicate that BiCGStab is by far the most efficient strategy, both in terms of the number of iterations and matrix-vector operations.