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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2014007778
pages 225-242

A MULTI-FIDELITY STOCHASTIC COLLOCATION METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA

Maziar Raissi
Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS: 3F2, Planetary Hall, Fairfax, Virginia 22030, USA
Padmanabhan Seshaiyer
Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS: 3F2, Planetary Hall, Fairfax, Virginia 22030, USA

ABSTRACT

Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte Carlo and the robustness of nonsampling ones like stochastic Galerkin to a great deal, have proved extremely useful in dealing with differential equations driven by random inputs. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient.