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

Nicolas Champagnat 1 Christophe Chipot 2 Erwan Faou 3, 4, *
* Corresponding author
1 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
2 SRSMC - Structure et Réactivité des Systèmes Moléculaires Complexes
3 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
4 IRMAR - Institut de Recherche Mathématique de Rennes
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.
Keywords : Least-squares approximation Monte Carlo methods high dimensional problems
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Nicolas Champagnat, Christophe Chipot, Erwan Faou. Reconciling alternate methods for the determination of charge distributions: A probabilistic approach to high-dimensional least-squares approximations. Journal of Mathematical Chemistry, Springer Verlag (Germany), 2011, Journal of Mathematical Chemistry, 49 (1), pp.296-324. ⟨10.1007/s10910-010-9740-0⟩. ⟨inria-00345411v2⟩

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