GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Computer simulation of liquids
Computer simulation of liquids
Matrix computations (3rd ed.)
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Experimental study of ILU preconditioners for indefinite matrices
Journal of Computational and Applied Mathematics
Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems
SIAM Journal on Scientific Computing
The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices
SIAM Journal on Matrix Analysis and Applications
Computational physics
Understanding Molecular Simulation
Understanding Molecular Simulation
On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix
SIAM Journal on Matrix Analysis and Applications
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing (Surveys and Tutorials in the Applied Mathematical Sciences)
Applied Numerical Mathematics
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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 incomplete LU decomposition with threshold and pivoting (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.