Journal of Mathematical Chemistry

, Volume 49, Issue 1, pp 296–324 | Cite as

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

Original Paper
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Abstract

We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. (Mol Phys 94: 881–895, 1998) 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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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.TOSCA project-teamINRIA Sophia Antipolis-MéditerranéeSophia Antipolis CedexFrance
  2. 2.Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.INRIA & École Normale Supérieure de Cachan BretagneBruzFrance

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