On the choice of subspace for iterative methods for linear discrete ill-posed problems

• Autorzy: Daniela Calvetti , Bryan Lewis , Lothar Reichel
• Języki publikacji: EN

Abstrakty

EN

Many iterative methods for the solution of linear discrete ill-posed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that when the desired solution is not smooth, it may be possible to determine meaningful approximate solutions with less computational work by not imposing this orthogonality condition.

Słowa kluczowe

EN

conjugate gradient method; ill-posed problems; minimal residual method

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2001
• Tom: 11
• Numer: 5
• Strony: 1069-1092

Daty

wydano

2001

Twórcy

autor

Daniela Calvetti

• Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, U.S.A.

autor

Bryan Lewis

• Department of Mathematics and Computer Science, Kent State University, Kent, OH 44242, U.S.A.

autor

Lothar Reichel

• Department of Mathematics and Computer Science, Kent State University, Kent, OH 44242, U.S.A.

Bibliografia

• Baart M.L. (1982): The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems. - IMA J. Numer. Anal., Vol.2, pp.241-247.
• Björck Å. (1996): Numerical Methods for Least Squares Problems. - Philadelphia: SIAM.
• Calvetti D., Lewis B. and Reichel L. (2000a): GMRES-type methods for inconsistent systems. - Lin. Alg. Appl., Vol.316, pp.157-169.
• Calvetti D., Lewis B. and Reichel L. (2000b): Restoration of images with spatially variant blur by the GMRES method, In: Advanced Signal Processing Algorithms, In: Architectures, and Implementations X (F.T. Luk, Ed.). - Proc. Society of Photo-Optical Instrumentation Engineers (SPIE), Vol.4116, Bellingham, WA: The International Society for Optical Engineering, pp.364-374.
• Calvetti D., Lewis B. and Reichel L. (2000c): An L-curve for the MINRES method, In: Advanced Signal Processing Algorithms, Architectures, and Implementations X (F.T. Luk, Ed.).- Proc. Society of Photo-Optical Instrumentation Engineers (SPIE), Vol.4116, Bellingham, WA: The International Society for Optical Engineering, pp.385-395.
• Calvetti D., Lewis B. and Reichel L. (2002a): GMRES, L-curves,and discrete ill-posed problems. - BIT, Vol.42, pp.44-65.
• Calvetti D., Lewis B. and Reichel L. (2002b): On the regularizing properties of the GMRES method. - Numer. Math., (to appear).
• Calvetti D., Reichel L. and Zhang Q. (1994a): An adaptive semiiterative method for symmetric semidefinite linear systems, In: Approximation and Computation (R.V.M. Zahar, Ed.). -Basel: Birkhauser, pp.77-96.
• Calvetti D., Reichel L. and Zhang Q. (1994b): Conjugate gradient algorithms for symmetric inconsistent linear systems, In: Proceedings of the Cornelius Lanczos International Centenary Conference (J.D. Brown, M.T. Chu, D.C. Ellison and R.J. Plemmons, Eds.).- Philadelphia: SIAM, pp.267-272.
• Calvetti D., Reichel L. and Zhang Q. (1999a): Iterative exponential filtering for large discrete ill-posed problems. - Numer. Math., Vol.83, pp.535-556.
• Calvetti D., Reichel L. and Zhang Q. (1999b): Iterative solution methods for large linear discrete ill-posedproblems. - Appl. Comp. Control Signals and Circuits, Vol.1, pp.313-367.
• Fischer B. (1996): Polynomial Based Iteration Methods for Symmetric Linear Systems. - New York: Wiley-Teubner.
• Hanke M. (1995): Conjugate Gradient Type Methods for Ill-Posed Problems. - Harlow: Longman.
• Hanke M. and Hansen P.C. (1993): Regularization methods for large-scale problems. - Surv. Math. Ind., Vol.3, pp.253-315.
• Hanke M. and Hochbruck M. (1993): A Chebyshev-like semiiteration for inconsistent linear systems. - Elec. Trans. Numer. Anal., Vol.1, pp.89-103.
• Hanke M. and Nagy J.G. (1998): Restoring images degraded by spatially-variant blur. - Inv. Problems, Vol.19, pp.1063-1082.
• Hansen P.C. (1994): Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems. - Numer. Algorithms, Vol.6, pp.1-35.
• Hansen P.C. (1998): Rank-Deficient and Discrete Ill-Posed Problems. - Philadelphia: SIAM.
• Nagy J.G. and O'Leary D.P. (1998): Restoring images degraded by spatially-variant blur. - SIAM J. Sci. Comput., Vol.19, pp.1063-1082.
• Paige C.C. and Saunders M.A. (1975): Solution of sparse indefinite systems of linear equations. - SIAM J. Numer. Anal., Vol.12, pp.617-629.
• Phillips D.L. (1962): A technique for the numerical solution of certain integral equations of the first kind. - J. ACM, Vol.9, pp.84-97.
• Saad Y. (1996): Iterative Methods for Sparse Linear Systems. -Boston: PWS.
• Saad Y. and Schultz M.H. (1986): GMRES: a generalized minimal residual method for solving nonsymmetric linear systems. - SIAM J. Sci. Stat. Comput., Vol.7, pp.856-869.