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Adaptive Stochastic Meshfree Methods for Optimal Control Problem Governed by Random Elliptic Equations

Adaptive Stochastic Meshfree Methods for Optimal Control Problem Governed by Random Elliptic Equations

Year:    2025

Author:    Liang Ge, Tongjun Sun, Wanfang Shen, Wenbin Liu

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 4 : pp. 813–839

Abstract

In this paper, a radial basis function method combined with the stochastic Galerkin method is considered to approximate elliptic optimal control problem with random coefficients. This radial basis function method is a meshfree approach for solving high dimensional random problem. Firstly, the optimality system of the model problem is derived and represented as a set of deterministic equations in high-dimensional parameter space by finite-dimensional noise assumption. Secondly, the approximation scheme is established by using finite element method for the physical space, and compactly supported radial basis functions for the parameter space. The radial basis functions lead to the sparsity of the stiff matrix with lower condition number. A residual type a posteriori error estimates with lower and upper bounds are derived for the state, co-state and control variables. An adaptive algorithm is developed to deal with the physical and parameter space, respectively. Numerical examples are presented to illustrate the theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2404-m2021-0289

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 4 : pp. 813–839

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    27

Keywords:    Radial basis function method Meshfree method Random elliptic equation A posteriori error estimates Stochastic Galerkin (SG) method Optimal control problem.

Author Details

Liang Ge

Tongjun Sun

Wanfang Shen

Wenbin Liu