[Paper Review] Current Status of Very-Large-Basis Hamiltonian Diagonalizations for Nuclear Physics

Calvin W. Johnson|arXiv (Cornell University)|Sep 20, 2018

Nuclear physics research studies28 references3 citations

TL;DR

This paper reviews the current state of very-large-basis Hamiltonian diagonalizations in nuclear physics, focusing on configuration-interaction methods and the interacting shell model. It details advances in computational techniques, basis state choices (M-scheme vs. J-scheme), and extrapolation strategies for convergence in ab initio calculations, highlighting progress toward matrix dimensions exceeding 20 billion using modern supercomputers and emerging machine learning approaches.

ABSTRACT

Today there are a plethora of many-body techniques for calculating nuclear wave functions and matrix elements. I review the status of that reliable workhorse, the interacting shell model, a.k.a. configuration-interaction methods, a.k.a. Hamiltonian diagonalization, and survey its advantages and disadvantages. With modern supercomputers one can tackle dimensions up to about 20 billion! I discuss how we got there and where we might go in the near future.

Motivation & Objective

• To assess the current capabilities and limitations of very-large-basis Hamiltonian diagonalizations in nuclear structure calculations.
• To compare the trade-offs between M-scheme and J-scheme basis representations in terms of computational efficiency and memory usage.
• To examine convergence and extrapolation techniques for ab initio calculations, especially in the no-core shell model (NCSM), as model space size increases.
• To evaluate emerging methods such as natural orbitals, symmetry-adapted bases, and machine learning for improving extrapolation and reducing computational cost.
• To identify future directions for advancing large-scale nuclear structure calculations using modern supercomputing and algorithmic innovations.

Proposed method

• Uses the configuration-interaction (CI) method to diagonalize the many-body nuclear Hamiltonian in a basis of Slater determinants or occupation-number representations.
• Employs the Lanczos algorithm to compute extremal eigenvalues of large sparse matrices, avoiding full diagonalization.
• Compares M-scheme bases (fixed M, or Jz) with J-scheme bases (fixed total J), analyzing trade-offs in basis dimension, matrix element sparsity, and computational cost.
• Applies exponential and parameterized extrapolation techniques (e.g., in Nmax and ℏΩ) to estimate infinite-space limits in no-core shell model (NCSM) calculations.
• Explores advanced basis optimizations such as natural orbitals and selected irreducible representations to improve convergence and reduce dimensionality.
• Investigates machine learning-based extrapolation methods to predict convergence behavior from limited data points, reducing reliance on brute-force basis expansion.

Experimental results

Research questions

• RQ1 How do different basis representations (M-scheme vs. J-scheme) affect the scalability and memory efficiency of large-scale Hamiltonian diagonalizations?
• RQ2 What are the most robust extrapolation techniques for estimating infinite-basis limits in ab initio nuclear structure calculations?
• RQ3 To what extent can machine learning improve the accuracy and efficiency of extrapolating nuclear energy spectra from finite model spaces?
• RQ4 How do improved single-particle orbitals, such as natural orbitals, enhance convergence in large-basis shell model calculations?
• RQ5 What are the computational and physical limitations of extending standard NCSM calculations to dimensions exceeding 20 billion?

Key findings

• Modern supercomputers now enable Hamiltonian diagonalizations with matrix dimensions exceeding 2 × 10^10, marking a significant advance from early shell model calculations.
• M-scheme bases, while requiring more states, allow efficient on-the-fly computation of matrix elements, reducing memory usage despite algorithmic complexity.
• J-scheme bases reduce basis dimensionality but require more complex algorithms and storage for matrix elements, increasing computational cost per element.
• Exponential extrapolations in Nmax and ℏΩ are less robust than newer methods that treat infrared and ultraviolet cutoffs as independent parameters.
• Combining Nmax and ℏΩ into physical parameters (e.g., wall effects) leads to more stable and reliable convergence estimates than traditional exponential fits.
• Machine learning-based extrapolation techniques show promise in predicting convergence behavior with limited data, though their broader applicability remains under investigation.