Abstract
Hierarchical (\(\mathcal{H}\))-matrices approximate full or sparse matrices using a hierarchical data sparse format. The corresponding \(\mathcal{H}\)-matrix arithmetic reduces the time complexity of the approximate \(\mathcal{H}\)-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 \(\mathcal{H}\)-matrix LU-factors as preconditioners in iterative methods. The experiment shows that the \(\mathcal{H}\)-matrix preconditioners are effective and speed up the convergence of iterative methods.
Chapter PDF
Similar content being viewed by others
References
Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Engineering Analysis with Boundary Elements 27, 405–422 (2003)
Börm, S., Grasedyck, L., Hackbush, W.: Hierarchical matrices. Lecture Notes No. 21. Max-Planck-Institute for Mathematics in the Sciences, Leipzig (2003)
Grasedyck, L., Hackbusch, W.: Construction and Arithmetics of H-matrices. Computing 70, 295–334 (2003)
Grasedyck, L., Kriemann, R., LeBorne, S.: Parallel Black Box Domain Decomposition Based H-LU Preconditioning. Mathematics of Computation, submitted (2005)
Gravvanis, G.: Explicit approximate inverse preconditioning techniques. Archives of Computational Methods in Engeneering 9 (2002)
Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices. Computing 62, 89–108 (1999)
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1999)
LeBorne, S.: Hierarchical matrix preconditioners for the Oseen equations. Comput. Vis. Sci. (2007)
LeBorne, S., Grasedyck, L.: H-matrix preconditioners in convection-dominated problems. SIAM J. Matrix Anal. Appl. 27, 1172–1183 (2006)
LeBorne, S., Grasedyck, L., Kriemann, R.: Domain-decomposition based H-LU preconditioners. LNCSE 55, 661–668 (2006)
Oliveira, S., Yang, F.: An Algebraic Approach for H-matrix Preconditioners. Computing, submmitted (2006)
Schaerer, C.E., Mathew, T., Sarkis, M.: Block iterative algorithms for the solution of parabolic optimal control problems. In: Daydé, M., Palma, J.M.L.M., Coutinho, Á.L.G.A., Pacitti, E., Lopes, J.C. (eds.) VECPAR 2006. LNCS, vol. 4395, pp. 452–465. Springer, Heidelberg (2007)
Yang, F.: H-matrix preconditioners and applications. PhD thesis. the University of Iowa
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Oliveira, S., Yang, F. (2007). Hierarchical-Matrix Preconditioners for Parabolic Optimal Control Problems. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2007. ICCS 2007. Lecture Notes in Computer Science, vol 4487. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72584-8_29
Download citation
DOI: https://doi.org/10.1007/978-3-540-72584-8_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72583-1
Online ISBN: 978-3-540-72584-8
eBook Packages: Computer ScienceComputer Science (R0)