Accelerating Eigenvalue Computation for Nuclear Structure Calculations via Perturbative Corrections

TL;DR

The paper introduces SPPC, a novel method that accelerates eigenvalue computations in nuclear structure calculations by combining hierarchical Hamiltonian partitioning with perturbative corrections, outperforming traditional iterative methods.

Contribution

The paper presents the SPPC method, which efficiently computes eigenvalues and eigenvectors by leveraging Hamiltonian hierarchy and perturbative corrections, improving over existing iterative techniques.

Findings

SPPC outperforms Lanczos, block Lanczos, and LOBPCG in efficiency.

Perturbative corrections improve eigenpair accuracy.

Method can be combined with other algorithms to prevent convergence issues.

Abstract

We present a new method for computing the lowest few eigenvalues and the corresponding eigenvectors of a nuclear many-body Hamiltonian represented in a truncated configuration interaction subspace, i.e., the no-core shell model (NCSM). The method uses the hierarchical structure of the NCSM Hamiltonian to partition the Hamiltonian as the sum of two matrices. The first matrix corresponds to the Hamiltonian represented in a small configuration space, whereas the second is viewed as the perturbation to the first matrix. Eigenvalues and eigenvectors of the first matrix can be computed efficiently. Perturbative corrections to the eigenvectors of the first matrix can be obtained from the solutions of a sequence of linear systems of equations defined in the small configuration space. These correction vectors can be combined with the approximate eigenvectors of the first matrix to construct a subspace from which more accurate approximations of the desired eigenpairs can be obtained. We call this method a Subspace Projection with Perturbative Corrections (SPPC) method. We show by numerical examples that the SPPC method can be more efficient than conventional iterative methods for solving large-scale eigenvalue problems such as the Lanczos, block Lanczos and the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The method can also be combined with other methods to avoid convergence stagnation.