Raul Fidel TEMPONE
Uncertainty Quantification with Multi-level and Multi-Index methods
Abstract: We start by recalling the Monte Carlo and Multi-level Monte Carlo (MLMC) methods for computing statistics of the solution of a Partial Differential Equation with random data. Then, we present the Multi-Index Monte Carlo (MIMC) and Multi-Index Stochastic Collocation (MISC) methods. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the MLMC method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically, thus yielding improved convergence rates. MISC is a deterministic combination technique that also uses mixed differences to achieve better complexity than MIMC, provided enough regularity. During the presentation, we will showcase the behavior of the numerical methods in applications, some of them arising in the context of Data Assimilation and Optimal Experimental Design.
Raul Tempone received his Ph.D. in 2002 in Numerical Analysis from the Royal Institute of Technology (Sweden). He was a post-doctoral fellow at the Institute for Computational and Engineering Sciences (ICES) at the University of Texas at Austin from 2003 to 2005. He is currently the head of the Stochastic Numerics Research Group at the King Abdullah University of Science and Technology (KAUST).
Raul Tempone’s research interests are in the mathematical foundation of computational science and engineering. More specifically, he has focused on a posteriori error approximation and related adaptive algorithms for numerical solutions of various differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. He is also interested in the development and analysis of efficient numerical methods for optimal control, uncertainty quantification and bayesian model calibration, validation and optimal experimental design. The areas of application he considers include, among others, engineering, chemistry, biology, physics as well as social science and computational finance.