High-quality preconditioning techniques for multi-length-scale symmetric positive definite matrices and their applications to the hybrid quantum Monte Carlo simulation of the Hubbard model

A hybrid quantum Monte Carlo (HQMC) simulation of the Hubbard model is a powerful tool for studying the electron interactions that characterize the fundamental properties of correlated materials. However, the HQMC simulation has been limited to hundreds of electrons because of the computational bottleneck of repeatedly solving the underlying multi-length-scale symmetric positive definite (SPD) linear systems of equations. In this dissertation, we design, analyze, and implement high-quality preconditioning techniques for the SPD linear systems and apply them to the HQMC simulation of systems consisting of thousands of electrons.;A variety of incomplete Cholesky (IC) preconditioners have been previously studied to solve the SPD linear systems using the preconditioned conjugate gradient (PCG) method. However, for ill-conditioned systems, these preconditioners are either very expensive or of low qualities. To address these issues, we propose a hybrid IC (HIC) preconditioner. We discuss algorithms to compute the preconditioner and introduce a new sparse matrix storage format which can efficiently accommodate the underlying data access pattern of the algorithm. We present numerical results to demonstrate the superior performance of the HIC preconditioner to solve the ill-conditioned linear systems.;We integrate the new preconditioning technique into the HQMC simulation and compute a number of physical observables of practical interest. We demonstrate that with the new preconditioning technique, the full simulation time scales linearly with respect to the number of electrons when the interactions between the electrons are moderate. As a result, we were able to address important questions concerning the magnetic and transport properties of materials composed of unprecedented thousands of electrons on a standard workstation.