dc.contributor.author Fox, Colin dc.contributor.author Parker, Albert E. dc.date.accessioned 2016-12-06T15:28:05Z dc.date.available 2016-12-06T15:28:05Z dc.date.issued 2014-02 dc.identifier.citation Fox C, Parker A, "Convergence in variance of Chebyshev accelerated Gibbs samplers," SIAM J. Sci. Comput. 2014 36(1), A124–A147 en_US dc.identifier.issn 1064-8275 dc.identifier.uri https://scholarworks.montana.edu/xmlui/handle/1/12331 dc.description.abstract A stochastic version of a stationary linear iterative solver may be designed to converge in distribution to a probability distribution with a specified mean $\mu$ and covariance matrix $A^{-1}$. A common example is Gibbs sampling applied to a multivariate Gaussian distribution which is a stochastic version of the Gauss--Seidel linear solver. The iteration operator that acts on the error in mean and covariance in the stochastic iteration is the same iteration operator that acts on the solution error in the linear solver, and thus both the stationary sampler and the stationary solver have the same error polynomial and geometric convergence rate. The polynomial acceleration techniques that are well known in numerical analysis for accelerating the linear solver may also be used to accelerate the stochastic iteration. We derive first-order and second-order Chebyshev polynomial acceleration for the stochastic iteration to accelerate convergence in the mean and covariance by mimicking the derivation for the linear solver. In particular, we show that the error polynomials are identical and hence so are the convergence rates. Thus, optimality of the Chebyshev accelerated solver implies optimality of the Chebyshev accelerated sampler. We give an algorithm for the stochastic version of the second-order Chebyshev accelerated SSOR (symmetric successive overrelaxation) iteration and provide numerical examples of sampling from multivariate Gaussian distributions to confirm that the desired convergence properties are achieved in finite precision. en_US dc.title Convergence in variance of Chebyshev accelerated Gibbs samplers en_US dc.type Article en_US mus.citation.extentfirstpage A124 en_US mus.citation.extentlastpage A147 en_US mus.citation.issue 1 en_US mus.citation.journaltitle SIAM Journal on Scientific Computing en_US mus.citation.volume 36 en_US mus.identifier.category Chemical & Material Sciences en_US mus.identifier.category Engineering & Computer Science en_US mus.identifier.category Life Sciences & Earth Sciences en_US mus.identifier.doi 10.1137/120900940 en_US mus.relation.college College of Agriculture en_US mus.relation.college College of Engineering en_US mus.relation.college College of Letters & Science en_US mus.relation.department Center for Biofilm Engineering. en_US mus.relation.department Chemical & Biological Engineering. en_US mus.relation.department Mathematical Sciences. en_US mus.relation.department Physics. en_US mus.relation.university Montana State University - Bozeman en_US mus.relation.researchgroup Center for Biofilm Engineering. en_US mus.data.thumbpage 13 en_US
﻿

### This item appears in the following Collection(s)

MSU uses DSpace software, copyright © 2002-2017  Duraspace. For library collections that are not accessible, we are committed to providing reasonable accommodations and timely access to users with disabilities. For assistance, please submit an accessibility request for library material.