Commun. Comput. Phys., 14 (2013), pp. 77-106.


A Stochastic Galerkin Method for Stochastic Control Problems

Hyung-Chun Lee 1, Jangwoon Lee 2*

1 Department of Mathematics, Ajou University, Suwon, Korea 443-749.
2 Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, USA.

Received 24 October 2011; Accepted (in revised version) 15 June 2012
Available online 18 October 2012
doi:10.4208/cicp.241011.150612a

Abstract

In an interdisciplinary field on mathematics and physics, we examine a physical problem, fluid flow in porous media, which is represented by a stochastic partial differential equation (SPDE). We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature. We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem. Finally, we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.

AMS subject classifications: 65M55, 65N30

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Key words: Stochastic Galerkin methods, stochastic partial differential equation, distributed control, finite element methods.

*Corresponding author.
Email: hclee@ajou.ac.kr (H.-C. Lee), llee3@umw.edu (J. Lee)
 

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