A stochastic density matrix approach to approximation of probability distributions and its application to nonlinear systems

Igor G. Vladimirov*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonor-mal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called stochastic density matrices in order to reflect an analogy with the quantum mechanical density matrices. The SDM approximation of a PDF satisfies the normalization condition and is nonnegative everywhere in contrast to the truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic structure, such as the Hermite polynomial and Fourier bases, the SDM approximation can be chosen so as to satisfy given moment specifications and can be optimized using a quadratic proximity criterion. We apply the SDM approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion processes governed by nonlinear stochastic differential equations. This leads to an ordinary differential equation for the SDM dynamics of the approximating PDF. As an example, we consider the Smoluchowski SDE on a multidimensional torus.

Original languageEnglish
Title of host publication2015 IEEE Conference on Control and Applications, CCA 2015 - Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1090-1095
Number of pages6
ISBN (Electronic)9781479977871
DOIs
Publication statusPublished - 4 Nov 2015
Externally publishedYes
EventIEEE Conference on Control and Applications, CCA 2015 - Sydney, Australia
Duration: 21 Sept 201523 Sept 2015

Publication series

Name2015 IEEE Conference on Control and Applications, CCA 2015 - Proceedings

Conference

ConferenceIEEE Conference on Control and Applications, CCA 2015
Country/TerritoryAustralia
CitySydney
Period21/09/1523/09/15

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