A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

ABSTRACT

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.

KEYWORDS:

• Meshless methods
• optimal control
• stochastic elliptic equations
• radial basis functions
• stochastic Galerkin method

2010 AMS SUBJECT CLASSIFICATIONS:

• 49J20
• 65N35

Acknowledgements

The authors are grateful to the referees for many important and profound suggestions on this paper. In particular, the substantial amendments for numerical examples in Section 6 are suggested by them, including the use of Matérn functions and the presentations of tables and figures.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11601466, 11671157, 91430213 the Nanhu Scholars Program for Young Scholars of XYNU; the Doctoral Scientific Research Startup Fund of Xinyang Normal University (2016).