Reconciling alternate methods for the determination of charge distributions: A probabilistic approach to high-dimensional least-squares approximations

TL;DR

This paper introduces a probabilistic Monte Carlo approach to high-dimensional least-squares problems, extending existing methods for distributed multipole analysis to improve computational tractability and robustness.

Contribution

It generalizes the SADM algorithm to broader least-squares problems, offering a new approximation technique suitable for ill-posed or high-dimensional cases.

Findings

The method converges under specific conditions.

It reduces computational cost for complex problems.

The approach is validated through theoretical analysis.

Abstract

We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. in [6] for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic potential. The method is mathematically extended to general least-squares problems and provides an alternative approximation method in cases where the original least-squares problem is computationally not tractable, either because of its ill-posedness or its high-dimensionality. The solution is approximated employing a Monte Carlo method that takes the average of a random variable defined as the solutions of random small least-squares problems drawn as subsystems of the original problem. The conditions that ensure convergence and consistency of the method are discussed, along with an analysis of the computational cost in specific instances.