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dc.contributor.authorFox, Colin
dc.contributor.authorParker, Albert E.
dc.identifier.citationFox C, Parker A, "Convergence in variance of Chebyshev accelerated Gibbs samplers," SIAM J. Sci. Comput. 2014 36(1), A124–A147en_US
dc.description.abstractA 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.titleConvergence in variance of Chebyshev accelerated Gibbs samplersen_US
mus.citation.journaltitleSIAM Journal on Scientific Computingen_US
mus.identifier.categoryChemical & Material Sciencesen_US
mus.identifier.categoryEngineering & Computer Scienceen_US
mus.identifier.categoryLife Sciences & Earth Sciencesen_US
mus.relation.collegeCollege of Agricultureen_US
mus.relation.collegeCollege of Engineeringen_US
mus.relation.collegeCollege of Letters & Scienceen_US
mus.relation.departmentCenter for Biofilm Engineering.en_US
mus.relation.departmentChemical & Biological Engineering.en_US
mus.relation.departmentMathematical Sciences.en_US
mus.relation.universityMontana State University - Bozemanen_US
mus.relation.researchgroupCenter for Biofilm Engineering.en_US

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