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Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

Published online by Cambridge University Press:  28 May 2015

Nary Kim  and
Hyung-Chun Lee
Nary Kim*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 443-749
Hyung-Chun Lee*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 443-749
*
Corresponding author. Email Address: gyory@naver.com
Corresponding author. Email Address: hclee@ajou.ac.kr
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Abstract

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

Keywords

49A2249B2265M5565N30Sparse grid collocationstochastic partial differential equationdistributed controlfinite element methods

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Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014 , pp. 166 - 188
Copyright
Copyright © Global-Science Press 2014

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