Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang
{"title":"分布椭圆型最优控制问题的鲁棒有限元离散化及求解方法","authors":"Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang","doi":"10.1515/cmam-2022-0138","DOIUrl":null,"url":null,"abstract":"Abstract We consider standard tracking-type, distributed elliptic optimal control problems with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} regularization, and their finite element discretization. We are investigating the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} error between the finite element approximation <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>ϱ</m:mi> <m:mo></m:mo> <m:mi>h</m:mi> </m:mrow> </m:msub> </m:math> u_{\\varrho h} of the state <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mi>ϱ</m:mi> </m:msub> </m:math> u_{\\varrho} and the desired state (target) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mi>u</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> \\overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϱ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> \\varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems\",\"authors\":\"Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang\",\"doi\":\"10.1515/cmam-2022-0138\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We consider standard tracking-type, distributed elliptic optimal control problems with <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} regularization, and their finite element discretization. We are investigating the <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} error between the finite element approximation <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>ϱ</m:mi> <m:mo></m:mo> <m:mi>h</m:mi> </m:mrow> </m:msub> </m:math> u_{\\\\varrho h} of the state <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>u</m:mi> <m:mi>ϱ</m:mi> </m:msub> </m:math> u_{\\\\varrho} and the desired state (target) <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mover accent=\\\"true\\\"> <m:mi>u</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> \\\\overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ϱ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> \\\\varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.\",\"PeriodicalId\":48751,\"journal\":{\"name\":\"Computational Methods in Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-02-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/cmam-2022-0138\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/cmam-2022-0138","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems
Abstract We consider standard tracking-type, distributed elliptic optimal control problems with L2 L^{2} regularization, and their finite element discretization. We are investigating the L2 L^{2} error between the finite element approximation uϱh u_{\varrho h} of the state uϱ u_{\varrho} and the desired state (target) u¯ \overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ=h4 \varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.
期刊介绍:
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.