OPTIMAL CONTROL FOR MULTISCALE ELLIPTIC EQUATIONS WITH ROUGH COEFFICIENTS

Yanping Chen, Xinliang Liu, Jiaoyan Zeng, Lei Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L∞ coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in composite materials and geophysics. We use one of the recently developed numerical homogenization techniques, the so-called Rough Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient resolution of the elliptic operator on the coarse scale. Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.
Original languageEnglish (US)
Pages (from-to)842-866
Number of pages25
JournalJOURNAL OF COMPUTATIONAL MATHEMATICS
Volume41
Issue number5
DOIs
StatePublished - May 2023

Bibliographical note

KAUST Repository Item: Exported on 2023-06-14
Acknowledgements: This work is supported by the State Key Program of National Natural Science Foundation of China (No. 11931003) and National Natural Science Foundation of China (Nos. 41974133, 11871339, 11861131004).

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