A Multilevel Stochastic Collocation Algorithm for Optimization of PDES with Uncertain Coefficients

TitleA Multilevel Stochastic Collocation Algorithm for Optimization of PDES with Uncertain Coefficients
Publication TypeConference Paper
Year of Publication2013
AuthorsKouri, DP
Conference NameSIAM Conference on Computational Science & Engineering
Date Published02/2013
Other Numbers ANL/MCS-P4037-0213
Abstract

In this work, we apply the MG/OPT framework to a multilevel in-sample-space discretization of optimization problems governed by PDEs with uncertain coefficients. The MG/OPT algorithm is a template for the application of multigrid to deterministic PDE optimization problems. We employ MG/OPT to exploit the hierarchical structure of sparse grids in order to formulate a multilevel stochastic collocation algorithm. The algorithm is provably first-order convergent under standard assumptions on the hierarchy of discretized objective functions as well as on the optimization routines used as pre- and post-smoothers. We present explicit bounds on the total number of PDE solves and an upper bound on the error for one V-cycle of the MG/OPT algorithm applied to a linear quadratic control problem. We provide numerical results that confirm the theoretical bound on the number of PDE solves and show a dramatic reduction in the total number of PDE solves required to solve these optimization problems when compared with standard optimization routines applied to the same problem.

PDFhttp://www.mcs.anl.gov/papers/P4037-0213.pdf