Abstract
We investigate the approximation properties of regularized solutions to discrete ill-posed least squares problems. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values. This is the discrete Picard condition. We illustrate the importance of this condition theoretically as well as experimentally.
Similar content being viewed by others
References
Å. Björck,A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT 28 (1988), 659–670.
Å. Björck & L. Eldén,Methods in numerical algebra for ill-posed problems, Report LITH-MAT-R33-1979, Dept. of Mathematics, Linköping University, 1979.
I. J. D. Craig & J. C. Brown,Inverse Problems in Astronomy, Adam Hilger, 1986.
U. Eckhardt & K. Mika,Numerical treatment of incorrectly posed problems — a case study; in J. Albrecht & L. Gollatz (Eds),Numerical Treatment of Integral Equations, Workshop on Numerical Treatment of Integral Equations, Oberwolfach, November 18–24, 1979, pp. 92–101, Birkhäuser, 1980.
M. Eiermann, I. Marek & W. Niethammer,On the solution of singular linear systems of algebraic equations by semiiterative methods, Numer. Math. 53 (1988), 265–283.
L. Eldén,Algorithms for regularization of ill-conditioned least-squares problems, BIT 17 (1977), 134–145.
L. Eldén,A weighted pseudoinverse, generalized singular values, and constrained least squares problems, BIT 22 (1982), 487–502.
L. Eldén,An algorithm for the regularization of ill-conditioned, banded least squares problems, SIAM J. Sci. Stat. Comput. 5 (1984), 237–254.
H. W. Engl & C. W. Groetsch (Eds.),Inverse and Ill-Posed Problems, Academic Press, 1987.
G. H. Golub, M. T. Heath & G. Wahba,Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21 (1979), 215–223.
J. Graves & P. M. Prenter,Numerical iterative filters applied to first kind Fredholm integral equations, Numer. Math. 30 (1978), 281–299.
C. W. Groetsch,The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, 1984.
P. C. Hansen,The truncated SVD as a method for regularization, BIT 27 (1987), 534–553.
P. C. Hansen,Computation of the singular value expansion, Computing 40 (1988), 185–199.
P. C. Hansen,Regularization, GSVD and truncated GSVD, BIT 29 (1989), 491–504.
P. C. Hansen,Perturbation bounds for discrete Tikhonov regularization, Inverse Problems 5 (1989), L41-L45.
P. C. Hansen,The discrete Picard condition for discrete ill-posed problems, CAM Report 89-22, Dept. of Mathematics, UCLA, 1989.
P. C. Hansen,Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Stat. Comput. 11 (1990), to appear.
R. J. Hanson,A numerical method for solving Fredholm integral equations of the first kind using singular values, SIAM J. Numer. Anal. 8 (1972), 883–890.
J. Larsen, H. Lund-Andersen & B. Krogsaa,Transient transport across the blood-retina barrier, Bulletin of Mathematical Biology 45 (1983), 749–758.
F. Natterer,The Mathematics of Computerized Tomography, Wiley, 1986.
F. Natterer,Numerical treatment of ill-posed problems; in G. Talenti (Ed.),Inverse Problems, pp. 142–167, Lecture Notes in Mathematics 1225, Springer, 1986.
D. P. O'Leary & B. W. Rust,Confidence intervals for inequality-constrained least squares problems, with applications to ill-posed problems, SIAM J. Sci. Stat. Comput. 7 (1986), 473–489.
D. P. O'Leary & J. A. Simmons,A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Stat. Comput. 2 (1981), 474–489.
C. C. Paige & M. A. Saunders,Towards a generalized singular value decomposition, SIAM J. Numer. Anal. 18 (1981), 398–405.
D. L. Phillips,A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9 (1962), 84–97.
A. N. Tikhonov,Solution of incorrectly formulated problems and the regularization method, Doklady Akad. Nauk SSSR 151 (1963), 501–504 = Soviet Math. 4 (1963), 1035–1038.
C. F. Van Loan,Generalizing the singular value decomposition, SIAM J. Numer. Anal. 13 (1976), 76–83.
A. van der Sluis & H. A. van der Vorst,SIRT and CG type methods for the iterative solution of sparse linear lest squares problems, Lin. Alg. Appl., 130 (1990), special issue on image processing.
J. M. Varah,A practical examination of some numerical methods for linear discrete ill-posed problems, SIAM Review 21 (1979), 100–111.
J. M. Varah,Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Stat. Comput. 4 (1983), 164–176.
Author information
Authors and Affiliations
Additional information
This work was carried out during a visit to Dept. of Mathematics, UCLA, and was supported by the Danish Natural Science Foundation, by the National Science Foundation under contract NSF-DMS87-14612, and by the Army Research Office under contract No. DAAL03-88-K-0085.
Rights and permissions
About this article
Cite this article
Hansen, P.C. The discrete picard condition for discrete ill-posed problems. BIT 30, 658–672 (1990). https://doi.org/10.1007/BF01933214
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01933214