A Fast Stochastic Galerkin Method for a Constrained Optimal Control Problem Governed by a Random Fractional Diffusion Equation

A Fast Stochastic Galerkin Method for a Constrained Optimal Control Problem Governed by a Random Fractional Diffusion Equation

Year:    2018

Author:    Ning Du, Wanfang Shen

Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 259–275

Abstract

We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tend to provide a better description for transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsened by the large number of random space dimensions to discretize the probability space. We approximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1612-m2016-0696

Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 259–275

Published online:    2018-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

Keywords:    Constrained optimal control Fractional diffusion Stochastic Galerkin method Fast Fourier transform Preconditioned Bi-Conjugate Gradient Stabilized method.

Author Details

Ning Du

Wanfang Shen

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