A Resolvent Quasi-Monte Carlo Method for Estimating the Minimum Eigenvalues Using the Error Balancing

There are iterative Monte Carlo (MC) methods that can be used for estimating the extreme eigenvalues of large dimensional matrices. The Power MC method allows for finding the approximate maximum eigenvalue of the considered matrix. In the case when we need to estimate the minimum eigenvalue, it is recommended to use the Resolvent MC method. The recent developments of quasi-random sequence generators and their successful application for solving large-scale problems motivate us to investigate the quasi-Monte Carlo (QMC) approaches for solving eigenvalue problems. In this work, we propose a Resolvent QMC algorithm to estimate the minimum eigenvalues of large-scale dimension symmetric matrices. To generate the quasi-random sequences we use BRODA’s Sobol Randomized Sequence Generator (RSG). Numerical experiments were done to investigate the balance between both errors - systematic and stochastic errors, which depend on the power of the resolvent matrix, the parameter controlling the convergence in the iteration process of the Resolvent MC/QMC method, and the number of realizations of the MC/QMC estimator. Numerical results show good scalability in the case of using Sobol sequences on GPU accelerators.