Convergence acceleration of polynomial chaos solutions via sequence transformation

We investigate convergence acceleration of the solution of stochastic differential equations characterized by their polynomial chaos expansions. Specifically, nonlinear sequence transformations are adapted to these expansions, viewed as a one-parameter family of functions with the parameter being th...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2014-04, Vol.271, p.167-184
Main Authors: Keshavarzzadeh, V., Ghanem, R.G., Masri, S.F., Aldraihem, O.J.
Format: Article
Language:English
Subjects:
Chaos theory
Convergence
Convergence acceleration
Mathematical analysis
Mathematical models
Nonlinearity
Polynomial chaos expansion
Polynomials
Sequence transformations
Stochastic Galerkin
Stochasticity
Transformations
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Summary:We investigate convergence acceleration of the solution of stochastic differential equations characterized by their polynomial chaos expansions. Specifically, nonlinear sequence transformations are adapted to these expansions, viewed as a one-parameter family of functions with the parameter being the polynomial degree of the expansion. These transformations can be generally viewed as nonlinear generalizations of Richardson Extrapolation and permit the estimation of coefficients in higher order expansions having knowledge of the coefficients in lower order ones. Stochastic Galerkin closure that typically characterizes the solution of such equations yields polynomial chaos representations that have the requisite analytical properties to ensure suitable convergence of these nonlinear sequence transformations. We investigate specifically Shanks and Levin transformations, and explore their properties in the context of a stochastic initial value problem and a stochastic elliptic problem.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2013.12.003