Efficient Algorithms for High-dimensional Eigenvalue Problems

Abstract

The eigenvalue problem is a traditional mathematical problem and has a wide applications. Although there are many algorithms and theories, it is still challenging to solve the leading eigenvalue problem of extreme high dimension. Full configuration interaction (FCI) problem in quantum chemistry is such a problem. This thesis tries to understand some existing algorithms of FCI problem and propose new efficient algorithms for the high-dimensional eigenvalue problem. In more details, we first establish a general framework of inexact power iteration and establish the convergence theorem of full configuration interaction quantum Monte Carlo (FCIQMC) and fast randomized iteration (FRI). Second, we reformulate the leading eigenvalue problem as an optimization problem, then compare the show the convergence of several coordinate descent methods (CDM) to solve the leading eigenvalue problem. Third, we propose a new efficient algorithm named Coordinate Descent Full Configuration Interaction (CDFCI) based on coordinate descent methods to solve the FCI problem, which produces some state-of-the-art results. Finally, we conduct various numerical experiments to fully test the algorithms.