  Quantum Monte Carlo (QMC) methods are often used to calculate properties of
many body quantum systems. The main cost of many QMC methods, for example the
variational Monte Carlo (VMC) method, is in constructing a sequence of Slater
matrices and computing the ratios of determinants for successive Slater
matrices. Recent work has improved the scaling of constructing Slater matrices
for insulators so that the cost of constructing Slater matrices in these
systems is now linear in the number of particles, whereas computing determinant
ratios remains cubic in the number of particles. With the long term aim of
simulating much larger systems, we improve the scaling of computing the
determinant ratios in the VMC method for simulating insulators by using
preconditioned iterative solvers.
  The main contribution of this paper is the development of a method to
efficiently compute for the Slater matrices a sequence of preconditioners that
make the iterative solver converge rapidly. This involves cheap preconditioner
updates, an effective reordering strategy, and a cheap method to monitor
instability of ILUTP preconditioners. Using the resulting preconditioned
iterative solvers to compute determinant ratios of consecutive Slater matrices
reduces the scaling of QMC algorithms from O(n^3) per sweep to roughly O(n^2),
where n is the number of particles, and a sweep is a sequence of n steps, each
attempting to move a distinct particle. We demonstrate experimentally that we
can achieve the improved scaling without increasing statistical errors. Our
results show that preconditioned iterative solvers can dramatically reduce the
cost of VMC for large(r) systems.
