Nonlinear multigrid for the solution of large-scale Riccati equations in low-rank and ℋ-matrix format

Numerical Linear Algebra with Applications

Volumn 15, Issue 9, 2008, Pages 779-807

  Grasedyck, L ^a  

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

Author keywords

Data sparse approximation; Hierarchical matrices; Low rank approximation; Multigrid method; Riccati equation

Indexed keywords

EID: 55749114945     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.606     Document Type: Article

References (37)

1. Arnold FW, Laub AJ. Generalized eigenproblem algorithms and software of algebraic Riccati equations. Pmceedings of the IEEE 1984; 72:1746-1754.
2. Casti JL. Dynamical Systems and their Applications: Linear Theory. Academic Press: New York, 1977.
3. Mehrmann VL. The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Lecture Notes in Control and inforniation Sciences, vol. 163. Springer: Heidelberg, 1991.
4. Hackbusch W. Multi-grid Methods and Applications (2nd edn). Springer: Berlin, 2003.
5. Trottenberg U, Oosterlee C, Schuller A. Multigrid. Academic Press: London, 2001.
6. Antoulas A, Sorensen D, Zhou Y. On the decay rate of Hankel singular values and related issues. Systems and Control Letters 2002; 46:323-342.
7. Grasedyck L. Existence of a low rank or ℋ-matrix approximant to the solution of a Sylvester equation. Numerical Linear Algebra with Applications 2004; 11:371-389.
8. Penzl T. Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Systems and Control Letters 2000; 40:139-144.
9. Kleinman DL. On an iterative technique for Riccati equations computation. IEEE Transactions on Automatic Control 1968; 13:114-115.
10. Grasedyck L, Hackbusch W. A multigrid method to solve large scale Sylvester equations. SIAM Journal on Matrix Analysis and Applications 2007; 29:870-894.
11. Penzl T. A multi-grid method for generalized Lyapunov equations. Technical Report SFB393/97-24, Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, FRG, 1997. (Available from: hitp://www.tu-chemnitz.de/ sfb393/ sfb97pr.html.)
12. Li JR, White J. Low rank solution of Lyapunov equations. SIAM Journal on Matrix Analysis and Applications 2002; 24:260-280.
13. Penzl T. A cyclic low rank Smith method for large sparse Lyapunov equations. SIAM Journal on Scientific Computing 2000; 21:1401-1418.
14. Stykel T. Generalized alternating direction implicit method for projected generalized Lyapunov equations. Proceedings in Applied Mathematics and Mechanics 2004; 4:686-687.
15. Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press: Cambridge, 2000.
16. Lezius R, Tröltzsch F. Theoretical and numerical aspects of controlled cooling of steel profiles. In Progress in Industrial Mathematics at ECMI94, Neunzert H (ed.). Wiley-Teubner: Leipzig, 1996; 380-388.
17. Benner B. Solving large-scale control problems. IEEE Control Systems Magazine 2004; 24:44-59.
18. Banks H, Kunisch K. The linear regulator problem for parabolic systems. SIAM Journal on Control and Optimization 1984; 22:684-696.
19. Gantmacher FR. Theory of Matrices. American Mathematical Society: New York, 2001.
20. Stéphanos C. Sur une extension du calculus des substitutions linéaires. Journal de Mathematiques Pures et Appliquees 1900; 6:73-128.
21. Yueh WC. Eigenvalues of several tridiagonal matrices. Applied Mathematics E-Notes 2005; 5:66-74.
22. Curtain R, Pritchard A. Infinite Dimensional Linear Systems Theory. Springer: New York, 1978.
23. Gibson JS. The Riccati integral equations for optimal control problems on Hilbert spaces. SIAM Journal on Control and Optimization 1979; 17:537-565.
24. Kwakernaak H, Sivan R. Linear Optimal Control Systems. Wiley-Interscience: New York, 1972.
25. Byers R. Numerical condition of the algebraic Riccati equation. Contemporary Mathematics 1985; 47:35-49.
26. Hackbusch W. Iterative Solution of Large Sparse Systems (2nd edn). Springer: Berlin, 2003.
27. Samarskii AA, Nikolaev ES. Numerical Methods for Grid Equations. Vol. II: Iterative Methods. Birkhäuser: Basel, 1989.
28. Reusken A. On maximum norm convergence of multigrid methods for two-point boundary value problems. SIAM Journal on Numerical Analysis 1992; 29:1569-1578.
29. Golub GH, Van Loan CH. Matrix Computations. Johns Hopkins University Press: London, 1996.
30. Grasedyck L, Hackbusch W. Construction and arithmetics of H-matrices. Computing 2003; 70:295-334.
31. Mehrmann VL, Tan E. Defect correction methods for the solution of algebraic Riccati equations. IEEE Transactions on Automatic Control 1988; 33:695-698.
32. Hackbusch W. A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing 1999; 62:89-108.
33. Grasedyck L, Hackbusch W, Khoromskij B. Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing 2003; 70:121-165.
34. Bad?́a JM, Benner P, Mayo R, Quintana-Ort?́ ES. Parallel solution of large-scale and sparse generalized algebraic Riccati equations. In Euro-Par 2006 Parallel Processing: 12th International Euro-Par Conference, Nagel WE, Walter WV, Lehner W (eds). Lecture Notes in Computer Science, vol. 4128. Springer: Berlin, 2006; 710-719.
35. Lu A, Wachspress L. Solution of Lyapunov equations by alternating direction implicit iteration. Computers and Mathematics with Applications 1991; 21:43-58.
36. Grasedyck L, Kriemann R, LeBorne S. Parallel black box domain decomposition based H-LU preconditioning. Technical Report 115, Max Planck Institute for Mathematics in the Sciences, Leipzig, 2005. (Available from: http://www.mis.mpg.de/preprints/2005/prepr2005 115.html.)
37. Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH. A supernodal approach to sparse partial pivoting. SIAM Journal on Matrix Analysis and Applications 1999; 20:720-755.