Uncertainty quantification in kinetic and hyperbolic problems

Multilveve Monte Carlo and Stochastic Collocation Methods

Max Gunzburger

Florida State University


We begin with a review of multilevel Monte Carlo (MLMC) methods for the solution of PDEs with random input data. We then consider stochastic collocation (SC) methods for this purpose. To alleviate the he curse of dimensionality, i.e., the explosive of the computational effort as the stochastic dimension increases, we propose and analyze a multilevel version of the SC method that, as is the case for MLMC methods, uses hierarchies of spatial approximations to reduce the overall computational complexity. In addition, our proposed approach utilizes, for approximation in stochastic space, a sequence of multi-dimensional interpolants of increasing fidelity which can then be used for approximating statistics of the solution as well as for building high-order surrogates featuring faster convergence rates. A rigorous convergence and computational cost analysis of the new multilevel stochastic collocation method is provided in the case of elliptic equations, demonstrating its advantages compared to standard single-level SC approximations as well as MLMC methods. Numerical results are provided that illustrate the theory and the effectiveness of the new multilevel method. (Joint work with Peter Jantsch, Aretha Teckentrup, Clayton Webster).