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Hierarchical-Matrix Preconditioners for Parabolic Optimal Control Problems
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Hierarchical-Matrix Preconditioners for Parabolic Optimal Control Problems

Suely Oliveira and Fang Yang
Computational Science – ICCS 2007, pp.221-228
Lecture Notes in Computer Science, Springer Berlin Heidelberg
2007
DOI: 10.1007/978-3-540-72584-8_29
url
https://doi.org/10.1007/978-3-540-72584-8_29View
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Abstract

Hierarchical (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document})-matrices approximate full or sparse matrices using a hierarchical data sparse format. The corresponding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-matrix arithmetic reduces the time complexity of the approximate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-matrix operators to almost optimal while maintains certain accuracy. In this paper, we represent a scheme to solve the saddle point system arising from the control of parabolic partial differential equations by using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-matrix LU-factors as preconditioners in iterative methods. The experiment shows that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-matrix preconditioners are effective and speed up the convergence of iterative methods.
hierarchical matrices multilevel methods parabolic optimal control problems

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