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

Abstract: Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as well as the need to develop efficient, scalable, stable and convergent computational methods for solving differential equations with random inputs. Stochastic Galerkin methods based on polynomial chaos expansions have shown superiority to other non-sampling and many sampling techniques. However, for complicated governing equations numerical implementations of stochastic Galerkin methods can become non-trivial. On the other hand, Monte Carlo and other traditional sampling methods, are straightforward to implement. However, they do not offer as fast convergence rates as stochastic Galerkin. Other numerical approaches are the stochastic collocation (SC) methods, which inherit both, the ease of implementation of Monte Carlo and the robustness of stochastic Galerkin to a great deal. 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.