Nested multigrid methods for time-periodic, parabolic optimal control problems

被引：21

作者：

Abbeloos, Dirk ^{[1
]}

Diehl, Moritz ^{[2
]}

Hinze, Michael ^{[3
]}

Vandewalle, Stefan ^{[1
]}

机构：

[1] Katholieke Univ Leuven, Dept Comp Sci, Celestijnenlaan 200, B-3001 Heverlee, Belgium

[2] Katholieke Univ Leuven, ESAT, Dept Elect Engn, B-3001 Heverlee, Belgium

[3] Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany

来源：

COMPUTING AND VISUALIZATION IN SCIENCE | 2011年 / 14卷 / 01期

关键词：

D O I：

10.1007/s00791-011-0158-4

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We present a nested multigrid method to optimize time-periodic, parabolic, partial differential equations (PDE). We consider a quadratic tracking objective with a linear parabolic PDE constraint. The first order optimality conditions, given by a coupled system of boundary value problems can be rewritten as an Fredholm integral equation of the second kind, which is solved by a multigrid of the second kind. The evaluation of the integral operator consists of solving sequentially a boundary value problem for respectively the state and the adjoints. Both problems are solved efficiently by a time-periodic space-time multigrid method.

引用

页码：27 / 38

页数：12

共 19 条

[1] Efficient numerical solution of parabolic optimization problems by finite element methods [J].

Becker, Roland ;

Meidner, Dominik ;

Vexler, Boris .

OPTIMIZATION METHODS & SOFTWARE, 2007, 22 (05) :813-833

[2] Multigrid methods for parabolic distributed optimal control problems [J].

Borzì, A .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2003, 157 (02) :365-382

[3] Multigrid Methods for PDE Optimization [J].

Borzi, Alfio ;

Schulz, Volker .

SIAM REVIEW, 2009, 51 (02) :361-395

[4] FAST NUMERICAL-SOLUTION OF TIME-PERIODIC PARABOLIC PROBLEMS BY A MULTIGRID METHOD [J].

HACKBUSCH, W .

SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING, 1981, 2 (02) :198-206

[5] FAST SOLVING OF PARABOLIC BOUNDARY CONTROL PROBLEMS [J].

HACKBUSCH, W .

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 1979, 17 (02) :231-244

[6]

HEMKER PW, 1981, MATH COMPUT, V36, P215, DOI 10.1090/S0025-5718-1981-0595054-2

[7]

Hinze M, 2009, MATH MODEL-THEOR APP, V23, P157, DOI 10.1007/978-1-4020-8839-1_3

[8] A SPACE-TIME MULTIGRID METHOD FOR PARABOLIC PARTIAL-DIFFERENTIAL EQUATIONS [J].

HORTON, G ;

VANDEWALLE, S .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 1995, 16 (04) :848-864

[9] Approximate Robust Optimization of Time-Periodic Stationary States with Application to Biochemical Processes [J].

Houska, Boris ;

Logist, Filip ;

Van Impe, Jan ;

Diehl, Moritz .

PROCEEDINGS OF THE 48TH IEEE CONFERENCE ON DECISION AND CONTROL, 2009 HELD JOINTLY WITH THE 2009 28TH CHINESE CONTROL CONFERENCE (CDC/CCC 2009), 2009, :6280-6285

[10]

Houska B, 2010, IEEE INTL CONF CONTR, P2172, DOI 10.1109/CCA.2010.5611288

← 1 2 →