A multigrid method to solve large scale Sylvester equations

SIAM Journal on Matrix Analysis and Applications

Volumn 29, Issue 3, 2007, Pages 870-894

  Grasedyck, Lars ^a   Hackbusch, Wolfgang ^a  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

Control problem; Fast solver; Low rank approximation; Lyapunov equation; Multigrid method; Riccati equation; Sylvester equation

Indexed keywords

LARGE SCALES; MULTIGRID (MG) METHOD; SYLVESTER EQUATIONS;

EID: 48249104699     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/040618102     Document Type: Article

References (21)

1. A. ANTOULAS, D. SORENSEN, AND Y. ZHOU, On the decay rate of Hankel singular values and related issues, Systems Control Lett., 46 (2002), pp. 323-342.
2. A. ANTOULAS AND D. SORENSEN, The Sylvester equation and approximate balanced reduction, Linear Algebra Appl., 351-352 (2002), pp. 671-700.
3. R. H. BARTELS AND G. W. STEWART, Solution of the matrix equation AX + XB = C, Comm. ACM, 15 (1972), pp. 820-826.
4. H. BANKS AND K. KUNISCH, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), pp. 684-696.
5. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins University Press, London, 1996.
6. L. GRASEDYCK, Existence of a low rank or H-matrix approximant to the solution of a Sylvester equation, Numer. Linear Algebra Appl., 11 (2004), pp. 371-389.
7. W. HACKBUSCH, Multi-Grid Methods and Applications, 2nd ed., Springer-Verlag, Berlin, 2003.
8. W. HACKBUSCH, Iterative Solution of Large Sparse Systems, 2nd ed., Springer-Verlag, Berlin, 2003.
9. W. HACKBUSCH, Elliptic Differential Equations. Theory and Numerical Treatment, 2nd ed., Springer-Verlag, Berlin, 2003.
10. W. HACKBUSCH AND S. SAUTER, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math., 75 (1997), pp. 447-472.
11. D. KLEINMAN, On an iterative technique for Riccati equations computation, IEEE Trans. Automat. Control, 13 (1968), pp. 114-115.
12. H. KWAKERNAAK AND R. SIVAN, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972.
13. I. LASIECKA AND R. TRIGGIANI, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University Press, Cambridge, 2000.
14. R. LEZIUS AND R. F. TRÖLTZSCH, Theoretical and numerical aspects of controlled cooling of steel profiles, in Progress in Industrial Mathematics at ECMI94, H. Neunzert, ed., Wiley-Teubner, Leipzig, 1996, pp. 380-388.
15. J. LI AND J. WHITE, LOW rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 260-280.
16. T. PENZL, Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case, Systems Control Lett., 40 (2000), pp. 139-144.
17. T. PENZL, A Multi-Grid Method for Generalized Lyapunov Equations, Technical report 24, SFB 393 at University Chemnitz, 1997.
18. T. PENZL, A cyclic low rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (2000), pp. 1401-1418.
19. J. RUGE AND K. STÜBEN, Efficient solution of finite difference and finite element equations by algebraic multigrid, in Multigrid Methods for Integral and Differential Equations, D. J. Paddon and H. Holstein, eds., Oxford University Press, NY, 1985, pp. 169-212.
20. D. HU AND L. REICHEL, Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl., 172 (1992), pp. 283-313.
21. U. TROTTENBERG, C. OOSTERLEE, AND A. SCHÜLLER, Multigrid, Academic Press, London, 2001.