A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost
In this paper, elliptic optimal control problems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-for...
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MDPI AG
2023-01-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/570 |
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| author | Xiaotong Chen Xiaoliang Song Zixuan Chen Lijun Xu |
| author_facet | Xiaotong Chen Xiaoliang Song Zixuan Chen Lijun Xu |
| author_sort | Xiaotong Chen |
| collection | DOAJ |
| description | In this paper, elliptic optimal control problems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-control cost and box constraints on the control are considered. To numerically solve the optimal control problems, we use the <i>First optimize, then discretize</i> approach. We focus on the inexact alternating direction method of multipliers (iADMM) and employ the standard piecewise linear finite element approach to discretize the subproblems in each iteration. However, in general, solving the subproblems is expensive, especially when the discretization is at a fine level. Motivated by the efficiency of the multigrid method for solving large-scale problems, we combine the multigrid strategy with the iADMM algorithm. Instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, to overcome the difficulty whereby the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm does not have a decoupled form, we apply nodal quadrature formulas to approximately discretize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-norm. Based on these strategies, an efficient multilevel heterogeneous ADMM (mhADMM) algorithm is proposed. The total error of the mhADMM consists of two parts: the discretization error resulting from the finite-element discretization and the iteration error resulting from solving the discretized subproblems. Both errors can be regarded as the error of inexactly solving infinite-dimensional subproblems. Thus, the mhADMM can be regarded as the iADMM in function space. Furthermore, theoretical results on the global convergence, as well as the iteration complexity results <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>o</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> for the mhADMM, are given. Numerical results show the efficiency of the mhADMM algorithm. |
| first_indexed | 2024-03-11T09:35:46Z |
| format | Article |
| id | doaj.art-a5461cc5a57346f59edb857ed32d2b39 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:35:46Z |
| publishDate | 2023-01-01 |
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| spelling | doaj.art-a5461cc5a57346f59edb857ed32d2b392023-11-16T17:21:29ZengMDPI AGMathematics2227-73902023-01-0111357010.3390/math11030570A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control CostXiaotong Chen0Xiaoliang Song1Zixuan Chen2Lijun Xu3School of Science, Dalian Maritime University, Dalian 116026, ChinaSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaCollege of Sciences, Northeastern University, Shenyang 110819, ChinaSchool of Science, Dalian Maritime University, Dalian 116026, ChinaIn this paper, elliptic optimal control problems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-control cost and box constraints on the control are considered. To numerically solve the optimal control problems, we use the <i>First optimize, then discretize</i> approach. We focus on the inexact alternating direction method of multipliers (iADMM) and employ the standard piecewise linear finite element approach to discretize the subproblems in each iteration. However, in general, solving the subproblems is expensive, especially when the discretization is at a fine level. Motivated by the efficiency of the multigrid method for solving large-scale problems, we combine the multigrid strategy with the iADMM algorithm. Instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, to overcome the difficulty whereby the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm does not have a decoupled form, we apply nodal quadrature formulas to approximately discretize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-norm. Based on these strategies, an efficient multilevel heterogeneous ADMM (mhADMM) algorithm is proposed. The total error of the mhADMM consists of two parts: the discretization error resulting from the finite-element discretization and the iteration error resulting from solving the discretized subproblems. Both errors can be regarded as the error of inexactly solving infinite-dimensional subproblems. Thus, the mhADMM can be regarded as the iADMM in function space. Furthermore, theoretical results on the global convergence, as well as the iteration complexity results <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>o</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> for the mhADMM, are given. Numerical results show the efficiency of the mhADMM algorithm.https://www.mdpi.com/2227-7390/11/3/570optimal control problemsADMMsparse regularizationmultilevel |
| spellingShingle | Xiaotong Chen Xiaoliang Song Zixuan Chen Lijun Xu A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost Mathematics optimal control problems ADMM sparse regularization multilevel |
| title | A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost |
| title_full | A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost |
| title_fullStr | A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost |
| title_full_unstemmed | A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost |
| title_short | A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with <i>L</i><sup>1</sup>-Control Cost |
| title_sort | multilevel heterogeneous admm algorithm for elliptic optimal control problems with i l i sup 1 sup control cost |
| topic | optimal control problems ADMM sparse regularization multilevel |
| url | https://www.mdpi.com/2227-7390/11/3/570 |
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