Pivoted Cauchy-like preconditioners for regularized solution of ill-posed problems

SIAM Journal of Scientific Computing

Volumn 21, Issue 1, 1999, Pages 88-110

  Kilmer, Misha E ^a,b   O'Leary, Dianne P ^c  

^a UNIVERSITY OF MARYLAND   (United States)

^b TUFTS UNIVERSITY   (United States)

^c NONE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; INTEGRAL EQUATIONS; ITERATIVE METHODS; LINEAR SYSTEMS; MATHEMATICAL TRANSFORMATIONS; MATRIX ALGEBRA; PROBLEM SOLVING; SIGNAL PROCESSING; VECTORS;

CAUCHY LIKE PRECONDITIONERS; CONJUGATE GRADIENT METHOD; ILL POSED PROBLEMS; REGULARIZATION; TOEPLITZ MATRIX;

LEAST SQUARES APPROXIMATIONS;

EID: 0033295054     PISSN: 10648275     EISSN: None     Source Type: None    
DOI: 10.1137/S1064827596308974     Document Type: Article

References (35)

1. A. BOJANCZYK, personal communication, 1996.
2. R. CHAN, J. NAGY, AND R. PLEMMONS, Circulant preconditioned Toeplitz least squares iterations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 80-97.
3. T. CHAN, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 766-771.
4. T. F. CHAN AND P. C. HANSEN, A lookahead Levinson algorithm for general Toeplitz systems, IEEE Proc. Signal Processing, 40 (1992), pp. 1079-1090.
5. R. D. FIERRO, G. H. GOLUB, P. C. HANSEN, AND D. P. O'LEARY, Regularization by truncated total least squares, in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, J. G. Lewis, ed., SIAM, Philadelphia, 1994, pp. 102-105.
6. T. FINCK, G. HEINIG, AND K. ROST, An inversion formula and fast algorithms for Cauchy-Vandermonde matrices, Linear Algebra Appl., 183 (1993), p. 179.
7. H. E. FLEMING, Equivalence of regularization and truncated iteration in the solution of ill-posed problems, Linear Algebra Appl., 130 (1990), pp. 133-150.
8. I. GOHBERG, T. KAILATH, AND V. OLSHEVSKY, Fast Gaussian elimination with partial pivoting of matrices with displacement structure, Math. Comp., 64 (1995), pp. 1557-1576.
9. W. GROETSCH, Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, MA, 1984.
10. M. GU, Stable and efficient algorithms for structured systems of linear equations, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 279-306.
11. M. H. GUTKNECHT AND M. HOCHBRUCK, Look-ahead Levinson and Schur Algorithms for Non-Hermitian Toeplitz Systems, Numer. Math., 70 (1995), pp. 181-227.
12. M. HANKE, Regularization with differential operators: An iterative approach, Numer. Funct. Anal. Optim., 13 (1992), pp. 523-540.
13. M. HANKE AND J. NAGY, Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques, Inverse Problems, 12 (1996), pp. 157-173.
14. M. HANKE, J. NAGY, AND R. PLEMMONS, Preconditioned iterative regularization for ill-posed problems, Numerical Linear Algebra, L. Reichel, A. Ruttan, and R. S. Varga, eds., de Gruyter, Berlin, 1993, pp. 141-163.
15. M. HANKE AND T. RAUS, A general heuristic for choosing the regularization parameter in ill-posed problems, SIAM J. Sci. Comput, 17 (1996), pp. 956-972.
16. P. C. HANSEN, The discrete Picard condition for discrete ill-posed problems, BIT, 30 (1990), pp. 658-672.
17. P. C. HANSEN, Rank Deficient and Discrete Ill-Posed Problems. Ph.D. thesis, Technical University of Denmark, Lyngby, Denmark, 1995.
18. P. C. HANSEN AND D. P. O'LEARY, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), pp. 1487-1503.
19. G. HEINIG, Inversion of generalized Cauchy matrices and other classes of structured matrices, in Linear Algebra in Signal Processing, IMA Vol. Math. Appl. 69, Springer-Verlag, New York, 1994, pp. 95-114.
20. G. HEINIG, personal communication. 1996.
21. G. HEINIG AND A. BOJANCZYK, Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices. I. Transformations, Linear Algebra Appl., 254 (1997), pp. 193-226.
22. M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Standards, 49 (1952), pp. 409-436.
23. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
24. T. KAILATH, S. KUNG, AND M. MORF, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl., 78 (1979), pp. 395-407.
25. T. KAILATH AND V. OLSHEVSKY, Displacement structure approach to discrete-trigonometric-transform based preconditioned of the G. Strang type and of T. Chan type, Calcolo, 33 (1996), pp. 191-208.
26. M. KILMER, Cauchy-Like Preconditioners for Two-Dimensional Ill-Posed Problems, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 777-799.
27. C. LANCZOS, Solution of systems of linear equations by minimized iterations, J. Res. Natl. Bur. Standards, 49 (1952), pp. 33-53.
28. D. L. PHILLIPS, A technique, for the numerical solution of certain integral equations of the first kind, J. ACM, 9 (1962), pp. 84-97.
29. G. W. STEWART AND J. G. SUN, Matrix Perturbation Theory, Academic Press, New York, 1990.
30. G. STRANG, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74 (1986), pp. 171-176.
31. D. SWEET AND R. BRENT, Error analysis of a fast partial pivoting method for structured matrices, Advanced Signal Processing Algorithms, Proc. SPIE, 2363 (1995), pp. 266-280.
32. D. SWEET, The use of pivoting to improve the numerical performance of Toeplitz matrix algorithms, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 468-493.
33. A. VAN DER SLUIS AND H. VAN DER VORST, The rate of convergence of conjugate gradients, Numer. Math, 48 (1986), pp. 543-560.
34. A. VAN DER SLUIS AND H. VAN DER VORST, Sirt- and CG-type methods for the iterative solution of sparse linear least-squares problems, Linear Algebra Appl., 130 (1990), pp. 257-302.
35. J. M. VARAH, Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 164-176.