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Over the last few years there have been dramatic advances in ourunderstanding of mathematical and computational models of complex systems inthe presence of uncertainty. This has led to a growth in the area ofuncertainty quantification as well as the need to develop efficient, scalable,stable and convergent computational methods for solving differential equationswith random inputs. Stochastic Galerkin methods based on polynomial chaosexpansions have shown superiority to other non-sampling and many samplingtechniques. However, for complicated governing equations numericalimplementations of stochastic Galerkin methods can become non-trivial. On theother hand, Monte Carlo and other traditional sampling methods, arestraightforward to implement. However, they do not offer as fast convergencerates as stochastic Galerkin. Other numerical approaches are the stochasticcollocation (SC) methods, which inherit both, the ease of implementation ofMonte Carlo and the robustness of stochastic Galerkin to a great deal. In thiswork we propose a novel enhancement to stochastic collocation methods usingdeterministic model reduction techniques. Linear parabolic partial differentialequations with random forcing terms are analysed. The input data are assumed tobe represented by a finite number of random variables. A rigorous convergenceanalysis, supported by numerical results, shows that the proposed technique isnot only reliable and robust but also efficient.

A multi-fidelity stochastic collocation method for parabolic PDEs with random input data