A riemannian optimization approach for computing low-rank solutions of lyapunov equations

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 5, 2010, Pages 2553-2579

  Vandereycken, Bart ^a   Vandewalle, Stefan ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Large scale equations; Low rank approximations; Lyapunov matrix equations; Numerical optimization on manifolds; Riemannian manifold

Indexed keywords

LARGE-SCALE EQUATIONS; LOW RANK APPROXIMATIONS; LYAPUNOV MATRIX EQUATIONS; NUMERICAL OPTIMIZATIONS; RIEMANNIAN MANIFOLD;

GEOMETRY; LYAPUNOV FUNCTIONS; OPTIMIZATION;

MATRIX ALGEBRA;

EID: 79251469735     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/090764566     Document Type: Article

References (47)

1. P.-A. Absil, C. Baker, and K. A. Gallivan, Trust-region methods on Riemannian manifolds, Found. Comput. Math., 7 (2007), pp. 303-330.
2. P.-A. Absil, M. Ishteva, L. De Lathauwer, and S. Van Huffel, A geometric Newton method for Oja's vector field, Neural Comput., 21 (2009), pp. 1415-1433.
3. P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.
4. A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005.
5. A. C. Antoulas, D. C. Sorensen, and Y. Zhou, On the decay rate of Hankel singular values and related issues, Systems Control Lett., 46 (2002), pp. 323-342.
6. C. Baker, P.-A. Absil, and K. Gallivan, GenRTR: A Riemannian Optimization Package, available online at http://www.math.fsu.edu/-cbaker/GenRTR.
7. R. H. Bartels and G. W. Stewart, Solution of the matrix equation AX +XB = C, Comm. ACM, 15 (1972), pp. 820-826.
8. P. Benner, Control Theory, Handbook of Linear Algebra, Chapman & Hall/CRC, Boca Raton, FL, 2006.
9. P. Benner and J. Saak, Efficient numerical solution of the LQR-problem for the heat equation, Proc. Appl. Math. Mech., 4 (2004), pp. 648-649.
10. M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137.
11. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Pure Appl. Math. 120, Academic Press, Orlando, 1986.
12. J. Boyle, M. D. Mihajlovic, and J. A. Scott, HSL MI20: An efficient AMG preconditioner for finite element problems in 3d, Internat. J. Numer. Methods Engrg., 82 (2010), pp. 64-98.
13. S. Burer and R. D. C. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program., 95 (2003), pp. 329-357.
14. K.-W. E. Chu, The solution of the matrix equation AXB ?CXD = E and (Y A? DZ,Y C ? BZ) = (E,F), Linear Algebra Appl., 93 (1987), pp. 93-105.
15. A. Edelman, T. A. Arias, and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 303-353.
16. H. Elman, D. Silvester, and A. Wathen, Finite Element and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, 2005.
17. G. H. Golub and C. F. V. Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1989.
18. 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.
19. L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 870-894.
20. S. Gugercin, D. C. Sorensen, and A. C. Antoulas, A modified low-rank Smith method for large-scale Lyapunov equations, Numer. Algorithms, 32 (2003), pp. 27-55.
21. U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer-Verlag, London, 1994.
22. U. Helmke and M. A. Shayman, Critical points of matrix least squares distance functions, Linear Algebra Appl., 215 (1995), pp. 1-19.
23. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
24. I. M. Jaimoukha and E. M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), pp. 227-251.
25. K. Jbilou and A. J. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl., 415 (2006), pp. 344-358. (Pubitemid 44304442)
26. M. Journée, F. Bach, P.-A. Absil, and R. Sepulchre, Low-rank optimization for semidefinite convex problems, SIAM J. Optim., 20 (2010), pp. 2327-2351.
27. O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454.
28. D. Kressner and C. Tobler., Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1688-1714.
29. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed., Academic Press, Orlando, 1985.
30. A. S. Lewis and J. Malick, Alternating projections on manifolds, Math. Oper. Res., 33 (2008), pp. 216-234.
31. J.-R. Li and J. White, Low-rank solution of Lyapunov equations, SIAM Rev., 46 (2004), pp. 693-713.
32. S. A. Miller and J. Malick, Newton methods for nonsmooth convex minimization: Connections among U-Lagrangian, Riemannian Newton and SQP methods, Math. Program., 104 (2005), pp. 609-633.
33. B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), pp. 17-32.
34. J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res., Springer-Verlag, New York, 1999.
35. R. Nong and D. C. Sorensen, A Parameter Free ADI-like Method for the Numerical Solution of Large Scale Lyapunov Equations, CAAM TR09-16, Computational and Applied Mathematics, Rice University, Houston, TX, 2009.
36. T. Penzl, Numerical solution of generalized Lyapunov equations, Adv. Comput. Math., 8 (1998), pp. 33-48.
37. T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (2000), pp. 1401-1418.
38. T. Penzl, Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case, Systems Control Lett., 40 (2000), pp. 139-144.
39. Y. Saad, Numerical solution of large Lyapunov equations, in Signal Processing, Scattering, Operator Theory, and Numerical Methods, M. A. Kaashoek, J. H. V. Schuppen, and A. C. M. Ran, eds., Birkhäuser Boston, Boston, MA, 1990, pp. 503-511.
40. J. Saak, H. Mena, and P. Benner, MESS (Matrix Equation Sparse Solver), available online at http://www-user.tu-chemnitz.de/?saak/Software/mess.php.
41. N. Scheerlinck, P. Verboven, J. D. Stigter, J. De Baerdemaeker, J. F. Van Impe, and B. M. Nicolai, A variance propagation algorithm for stochastic heat and mass transfer problems in food processes, Internat. J. Numer. Methods Engrg., 51 (2001), pp. 961-983.
42. V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29 (2007), pp. 1268-1288.
43. D. C. Sorensen and Y. Zhou, Bounds on Eigenvalue Decay Rates and Sensitivity of Solutions to Lyapunov Equations, CAAM TR02-07, Computational and Applied Mathematics, Rice University, Houston, TX, 2002.
44. T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), pp. 626-637.
45. G. W. Stewart, Matrix Algorithms. Volume II: Eigensystems, SIAM, Philadelphia, 2001.
46. P. L. Toint, Towards an efficient sparsity exploiting Newton method for minimization, in Sparse Matrices and Their Uses, Academic Press, London, New York, 1981, pp. 57-88.
47. C. F. Van Loan, The ubiquitous Kronecker product, J. Comput. Appl. Math., 123 (2000), pp. 85-100.