Accelerating Eigenvalue Computation for Nuclear Structure Calculations via Perturbative Corrections

TL;DR

The paper introduces SPPC, a perturbation-informed subspace projection method for accelerating the computation of the lowest eigenpairs of large, sparse nuclear Hamiltonians encountered in the no-core shell model. By splitting the Hamiltonian as $H=H_0+V$ with $H_0$ drawn from a smaller configuration space, SPPC builds a low-dimensional subspace from perturbative corrections to the $H_0$-eigenvectors and computes eigenpairs via Rayleigh-Ritz projections. The approach often outperforms traditional solvers in early iterations and can be hybridized with methods like RMM-DIIS to mitigate stagnation, yielding robust, fast convergence for computing a few lowest eigenpairs. The work demonstrates substantial reductions in sparse-matrix-vector products (SpMVs) across several light nuclei and emphasizes practical integration with existing nuclear-structure codes for scalable high-performance computing.

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.

Figures (6)

• Figure 1: (a) The Hamiltonian matrix $H$ of ${}^{12}$C constructed by the CI method with a truncation parameter $N_{\max}=4$, using the nucleon-nucleon interaction. The leading submatrix $\hat{H}_0$ of $H$ that is around $63$ times smaller is equivalent to the Hamiltonian matrix constructed by the CI method with a truncation parameter $N_{\max}=2$. (b) Illustration of the eigenvector localization observed in the Hamiltonian matrix. The leading components of the eigenvector corresponding to the lowest eigenvalue of $H$ are several magnitudes larger than the tailing components.
• Figure 2: The convergence of the algorithms for computing the lowest eigenpair of the Hamiltonians matrices (${}^{12}$C on the left and ${}^{6}$Li on the right). One iteration equals one SpMV for all algorithms.
• Figure 3: The subspace angle between the correction vector at iteration $p$ and the subspace spanned by the previous correction vectors from iteration $1$ to $p-1$.
• Figure 4: Algorithm \ref{['alg:SPPC_k']} to compute the first $5$ eigenpairs of the ${}^{6}$Li Hamiltonian, one by one. The left plot shows the relative residual norm, and the right plot shows the approximated eigenvalues in comparison with the true eigenvalues.
• Figure 5: Algorithm \ref{['alg:SPPC']} to compute the first $5$ eigenpairs of the ${}^{6}$Li Hamiltonian. The left plot shows the relative residual norm, and the right plot shows the approximated eigenvalues in comparison with the true eigenvalues.