High Performance LOBPCG Method for Solving Multiple Eigenvalues of Hubbard Model: Efficiency of Communication Avoiding Neumann Expansion Preconditioner.

The exact diagonalization method is a high accuracy numerical approach for solving the Hubbard model of a system of electrons with strong correlation. The method solves for the eigenvalues and eigenvectors of the Hamiltonian matrix derived from the Hubbard model. Since the Hamiltonian is a huge sparse symmetric matrix, it was expected that the LOBPCG method with an appropriate preconditioner could be used to solve the problem in a short time. This turned out to be the case as the LOBPCG method with a suitable preconditioner succeeded in solving the ground state (the smallest eigenvalue and its corresponding eigenvector) of the Hamiltonian. In order to solve for multiple eigenvalues of the Hamiltonian in a short time, we use a preconditioner based on the Neumann expansion which uses approximate eigenvalues and eigenvectors given by LOBPCG iteration. We apply a communication avoiding strategy, which was developed considering the physical properties of the Hubbard model, to the preconditioner. Our numerical experiments on two parallel computers show that the LOBPCG method coupled with the Neumann preconditioner and the communication avoiding strategy improves convergence and achieves excellent scalability when solving for multiple eigenvalues.