Abstract:  Development of efficient numerical methods for the solution of problems with high-dimensional stochastic  inputs has been a subject of active research in computational sciences and engineering. This is motivated by the need to reduce the computational cost of Uncertainty Quantification (UQ). In this talk, I will discuss two recently developed UQ algorithms that are particularly suitable for high dimensional large scale simulations. These methods are:

(1) Multifidelity stochastic collocation. The method combines the computational efficiency of low-fidelity models with the high accuracy of expensive high-fidelity models. The method can be useful when the computational resources are limited. And  it is non-intrusive and applicable to black-box simulation tools.

(2) Localized polynomial chaos expansion for stochastic PDEs with high dimensional random inputs. The method employs  domain decomposition technique to approximate the problem locally. The subdomain problems are solved independently and in much lower random dimensions. Accurate global solution can then be obtained by enforcing the correct statistical dependence. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient.