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Volume 41, Issue 5
Optimal Control for Multiscale Elliptic Equations with Rough Coefficients

Yanping Chen, Xinliang Liu, Jiaoyan Zeng & Lei Zhang

DOI: 10.4208/jcm.2112-m2021-0123

J. Comp. Math., 41 (2023), pp. 841-865.

Published online: 2023-05

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  • 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.

  • Keywords

Optimal control, Rough coefficients, Multiscale elliptic equations, Numerical homogenization, Rough polyharmonic splines, Iterative algorithm.

  • AMS Subject Headings

65M06, 65M12, 65M15, 65M55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yanpingchen@scnu.edu.cn (Yanping Chen)

xinliang.liu@kaust.edu.sa (Xinliang Liu)

zengjiaoyan2013@163.com (Jiaoyan Zeng)

lzhang2012@sjtu.edu.cn (Lei Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-841, author = {Chen , YanpingLiu , XinliangZeng , Jiaoyan and Zhang , Lei}, title = {Optimal Control for Multiscale Elliptic Equations with Rough Coefficients}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {841--865}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2112-m2021-0123}, url = {http://global-sci.org/intro/article_detail/jcm/21676.html} }
TY - JOUR T1 - Optimal Control for Multiscale Elliptic Equations with Rough Coefficients AU - Chen , Yanping AU - Liu , Xinliang AU - Zeng , Jiaoyan AU - Zhang , Lei JO - Journal of Computational Mathematics VL - 5 SP - 841 EP - 865 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2112-m2021-0123 UR - https://global-sci.org/intro/article_detail/jcm/21676.html KW - Optimal control, Rough coefficients, Multiscale elliptic equations, Numerical homogenization, Rough polyharmonic splines, Iterative algorithm. AB -

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.

Chen , YanpingLiu , XinliangZeng , Jiaoyan and Zhang , Lei. (2023). Optimal Control for Multiscale Elliptic Equations with Rough Coefficients. Journal of Computational Mathematics. 41 (5). 841-865. doi:10.4208/jcm.2112-m2021-0123
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