Volume 4, Issue 2
Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

Nary Kim & Hyung-Chun Lee

East Asian J. Appl. Math., 4 (2014), pp. 166-188.

Published online: 2018-02

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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

Sparse grid collocation, stochastic partial differential equation, distributed control, finite element methods.

  • AMS Subject Headings

49A22, 49B22, 65M55, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-4-166, author = {}, title = {Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {4}, number = {2}, pages = {166--188}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.041013.180314a}, url = {http://global-sci.org/intro/article_detail/eajam/10830.html} }
TY - JOUR T1 - Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs JO - East Asian Journal on Applied Mathematics VL - 2 SP - 166 EP - 188 PY - 2018 DA - 2018/02 SN - 4 DO - http://doi.org/10.4208/eajam.041013.180314a UR - https://global-sci.org/intro/article_detail/eajam/10830.html KW - Sparse grid collocation, stochastic partial differential equation, distributed control, finite element methods. AB -

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.

Nary Kim & Hyung-Chun Lee. (1970). Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs. East Asian Journal on Applied Mathematics. 4 (2). 166-188. doi:10.4208/eajam.041013.180314a
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