The discrete picard condition for discrete ill-posed problems Part II Numerical Mathematics Received: 15 August 1989 Revised: 15 February 1990 DOI :
10.1007/BF01933214

Cite this article as: Hansen, P.C. BIT (1990) 30: 658. doi:10.1007/BF01933214
115
Citations
1.2k
Downloads
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.

AMS Subject classification 65F30 65F20

Key words Ill-posed problems Tikhonov regularization discrete Picard condition generalized SVD 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.

References [1]

Å. Björck,

A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations , BIT 28 (1988), 659–670.

Google Scholar [2]

Å. Björck & L. Eldén,Methods in numerical algebra for ill-posed problems , Report LITH-MAT-R33-1979, Dept. of Mathematics, Linköping University, 1979.

[3]

I. J. D. Craig & J. C. Brown,Inverse Problems in Astronomy , Adam Hilger, 1986.

[4]

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.

[5]

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.

Google Scholar [6]

L. Eldén,

Algorithms for regularization of ill-conditioned least-squares problems , BIT 17 (1977), 134–145.

Google Scholar [7]

L. Eldén,

A weighted pseudoinverse, generalized singular values, and constrained least squares problems , BIT 22 (1982), 487–502.

Google Scholar [8]

L. Eldén,

An algorithm for the regularization of ill-conditioned, banded least squares problems , SIAM J. Sci. Stat. Comput. 5 (1984), 237–254.

Google Scholar [9]

H. W. Engl & C. W. Groetsch (Eds.),Inverse and Ill-Posed Problems , Academic Press, 1987.

[10]

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.

Google Scholar [11]

J. Graves & P. M. Prenter,

Numerical iterative filters applied to first kind Fredholm integral equations , Numer. Math. 30 (1978), 281–299.

Google Scholar [12]

C. W. Groetsch,The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind , Pitman, 1984.

[13]

P. C. Hansen,

The truncated SVD as a method for regularization , BIT 27 (1987), 534–553.

Google Scholar [14]

P. C. Hansen,

Computation of the singular value expansion , Computing 40 (1988), 185–199.

Google Scholar [15]

P. C. Hansen,

Regularization, GSVD and truncated GSVD , BIT 29 (1989), 491–504.

Google Scholar [16]

P. C. Hansen,

Perturbation bounds for discrete Tikhonov regularization , Inverse Problems 5 (1989), L41-L45.

Google Scholar [17]

P. C. Hansen,The discrete Picard condition for discrete ill-posed problems , CAM Report 89-22, Dept. of Mathematics, UCLA, 1989.

[18]

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.

[19]

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.

Google Scholar [20]

J. Larsen, H. Lund-Andersen & B. Krogsaa,

Transient transport across the blood-retina barrier , Bulletin of Mathematical Biology 45 (1983), 749–758.

Google Scholar [21]

F. Natterer,The Mathematics of Computerized Tomography , Wiley, 1986.

[22]

F. Natterer,Numerical treatment of ill-posed problems; in G. Talenti (Ed.),Inverse Problems , pp. 142–167, Lecture Notes in Mathematics 1225, Springer, 1986.

[23]

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.

Google Scholar [24]

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.

Google Scholar [25]

C. C. Paige & M. A. Saunders,

Towards a generalized singular value decomposition , SIAM J. Numer. Anal. 18 (1981), 398–405.

Google Scholar [26]

D. L. Phillips,

A technique for the numerical solution of certain integral equations of the first kind , J. ACM 9 (1962), 84–97.

Google Scholar [27]

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.

Google Scholar [28]

C. F. Van Loan,

Generalizing the singular value decomposition , SIAM J. Numer. Anal. 13 (1976), 76–83.

Google Scholar [29]

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.

[30]

J. M. Varah,

A practical examination of some numerical methods for linear discrete ill-posed problems , SIAM Review 21 (1979), 100–111.

Google Scholar [31]

J. M. Varah,

Pitfalls in the numerical solution of linear ill-posed problems , SIAM J. Sci. Stat. Comput. 4 (1983), 164–176.

Google Scholar Authors and Affiliations 1. UNI•C Technical University of Denmark Lyngby Denmark