# Iterative approaches to the self-consistent nuclear energy density functional problem. Heavy ball dynamics and potential preconditioning

## TL;DR

This paper introduces two algorithmic improvements, heavy-ball dynamics and potential preconditioning, that significantly accelerate the convergence of self-consistent nuclear energy density functional calculations, reducing CPU time and eliminating manual parameter tuning.

## Contribution

The paper proposes two easily implementable, nearly parameter-free algorithms that enhance the efficiency of solving the non-linear self-consistency problem in nuclear EDF methods.

## Key findings

- Heavy-ball dynamics and potential preconditioning reduce CPU time for convergence.
- Algorithms require minimal manual tuning, improving robustness.
- Demonstrated effectiveness on 3D coordinate-space HFB calculations.

## 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.