Iterative approaches to the self-consistent nuclear energy density functional problem

Abstract

Large-scale applications of energy density functional (EDF) methods depend on fast and reliable algorithms to solve the associated non-linear self-consistency problem. When dealing with large single-particle variational spaces, existing solvers can become very slow, and their performance dependent on manual fine-tuning of numerical parameters. In addition, convergence can sensitively depend on particularities of the EDF's parametrisation under consideration. Using the widely-used Skyrme EDF as an example, we investigate the impact of the parametrisation of the EDF, both in terms of the operator structures present and the size of coupling constants, on the convergence of numerical solvers. We focus on two aspects of the self-consistency cycle, which are the diagonalisation of a fixed single-particle Hamiltonian on one hand and the evolution of the mean-field densities and potentials on the other. Throughout the article we use a coordinate-space representation, for which the behaviour of algorithms can be straightforwardly analysed. We propose two algorithmic improvements that are easily implementable in existing solvers, heavy-ball dynamics and potential preconditioning. We demonstrate that these methods can be made virtually parameter-free, requiring no manual fine-tuning to achieve near-optimal performance except for isolated cases. The combination of both methods decreases substantially the CPU time required to obtain converged results. The improvements are illustrated for the MOCCa code that solves the self-consistent HFB problem in a 3d coordinate space representation for parametrisations of the standard Skyrme EDF at next-to-leading order in gradients and its extension to next-to-next-to-leading order.

Finite-temperature mean-field approximations for shell model Hamiltonians: the code HF-SHELL

“…Here we employ the heavy-ball optimization method of Ref. [2], a simple extension of the original method of Ref. [34], which we describe in Sect.…”

“…The heavy-ball algorithm was recently proposed [2] to iteratively diagonalize the single-particle Hamiltonian ĥ. We assume a set of orthonormal single-particle wavefunctions |ψ…”

“…Numerically, the code offers a robust self-consistency cycle through the use of the heavy-ball method of Ref. [2], which is similar in spirit to the gradient method of Refs. [25,27].…”

Variational approximations to exact solutions in shell-model valence spaces: Calcium isotopes in the pf shell

“…Nevertheless, we will compare the results obtained by PGCM schemes using both HFB and PNVAP underlying states. In both cases, the gradient method [43,44] complemented by a momentum term [45] is used to solve the non-linear equations at hand.…”

Search for octupole-deformed actinium isotopes using resonance ionization spectroscopy

“…The calculations with SLy5s1 on the other hand were carried out with the MOCCa code, that represents the wave functions in three-dimensional (3D) coordinate space. The code allows for the simultaneous breaking of reflection and time-reversal symmetry, and is equipped with algorithms that enable it to reliably find the one-or two-quasiparticle configuration with the lowest energy [84], and alleviates the numerical burden of symmetry-broken 3D calculations [85]. To assess the effect of octupole deformation, results obtained with SLy5s1 from full calculations allowing for reflectionasymmetric shapes and from restricted calculations that impose reflection symmetry are presented.…”