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Volume 36, Issue 5
Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients

Liang Ge, Ningning Yan, Lianhai Wang, Wenbin Liu & Danping Yang

DOI: 10.4208/jcm.1703-m2015-0433

J. Comp. Math., 36 (2018), pp. 644-660.

Published online: 2018-06

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  • Abstract

In this paper, we investigate heterogeneous multiscale method (HMM) for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well-known Lions' Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both $L^2$ and $H^1$ norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

  • Keywords

Constrained convex optimal control, Heterogeneous multiscale finite element, A priori error estimate, Elliptic equations with highly oscillatory coefficients.

  • AMS Subject Headings

65N06, 65B99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

geliang@sdu.edu.cn (Liang Ge)

ynn@amss.ac.cn (Ningning Yan)

wanglh@sdas.org (Lianhai Wang)

W.B.Liu@kent.ac.uk (Wenbin Liu)

dpyang@math.ecnu.edu.cn (Danping Yang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-644, author = {Ge , LiangYan , NingningWang , LianhaiLiu , Wenbin and Yang , Danping}, title = {Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {5}, pages = {644--660}, abstract = {

In this paper, we investigate heterogeneous multiscale method (HMM) for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well-known Lions' Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both $L^2$ and $H^1$ norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1703-m2015-0433}, url = {http://global-sci.org/intro/article_detail/jcm/12450.html} }
TY - JOUR T1 - Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients AU - Ge , Liang AU - Yan , Ningning AU - Wang , Lianhai AU - Liu , Wenbin AU - Yang , Danping JO - Journal of Computational Mathematics VL - 5 SP - 644 EP - 660 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1703-m2015-0433 UR - https://global-sci.org/intro/article_detail/jcm/12450.html KW - Constrained convex optimal control, Heterogeneous multiscale finite element, A priori error estimate, Elliptic equations with highly oscillatory coefficients. AB -

In this paper, we investigate heterogeneous multiscale method (HMM) for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well-known Lions' Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both $L^2$ and $H^1$ norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

Ge , LiangYan , NingningWang , LianhaiLiu , Wenbin and Yang , Danping. (2018). Heterogeneous Multiscale Method for Optimal Control Problem Governed by Elliptic Equations with Highly Oscillatory Coefficients. Journal of Computational Mathematics. 36 (5). 644-660. doi:10.4208/jcm.1703-m2015-0433
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