Numerical Algorithms

, Volume 6, Issue 1, pp 1–35

# REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

• Per Christian Hansen
Article

## Abstract

The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero. Such problems typically arise in connection with discretization of Fredholm integral equations of the first kind, and similar ill-posed problems. Some form of regularization is always required in order to compute a stabilized solution to discrete ill-posed problems. The purpose of REGULARIZATION TOOLS is to provide the user with easy-to-use routines, based on numerical robust and efficient algorithms, for doing experiments with regularization of discrete ill-posed problems. By means of this package, the user can experiment with different regularization strategies, compare them, and draw conclusions from these experiments that would otherwise require a major programming effert. For discrete ill-posed problems, which are indeed difficult to treat numerically, such an approach is certainly superior to a single black-box routine. This paper describes the underlying theory gives an overview of the package; a complete manual is also available.

## Keywords

Regularization ill-posed problems ill-conditioning, Matlab Subject classification AMS(MOS) 65F30 65F20

## References

1. [1]
R.C. Allen, Jr., W.R. Boland, V. Faber and G.M. Wing, Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind, J. Math. Anal. Appl. 109 (1985) 564–590.Google Scholar
2. [2]
R.S. Anderssen and P.M. Prenter, A formal comparison of methods proposed for the numerical solution of first kind integral equations, J. Austral. Math. Soc. (Series B) 22 (1981) 488–500.Google Scholar
3. [3]
C.T.H. Baker, Expansion methods, chapter 7 in [18].Google Scholar
4. [4]
C.T.H. Baker,The Numerical Treatment of Integral Equations (Clarendon Press, Oxford, 1977).Google Scholar
5. [5]
C.T.H. Baker and M.J. Miller (eds.),Treatment of Integral Equations by Numerical Methods (Academic Press, New York, 1982).Google Scholar
6. [6]
D.M. Bates, M.J. Lindstrom, G. Wahba and B.S. Yandell, GCVPACK — routines for generalized cross validation, Commun. Statist.-Simul. 16 (1987) 263–297.Google Scholar
7. [7]
M. Bertero, C. De Mol and E.R. Pike, Linear inverse problems with discrete data: II. Stability and regularization, Inverse Problems 4 (1988) 573–594.Google Scholar
8. [8]
Å. Björck, A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT 28 (1988) 659–670.Google Scholar
9. [9]
Å. Björck, Least squares methods, in:Handbook of Numerical Analysis, Vol. 1, eds. P.G. Ciarlet and J.L. Lions (Elsevier, Amsterdam, 1990).Google Scholar
10. [10]
Å. Björck and L. Eldén, Methods in numerical algebra for ill-posed problems, Report LiTH-MAT-R33-1979, Dept. of Mathematics, Linköping University (1979).Google Scholar
11. [11]
H. Brakhage, On ill-possed problems and the method of conjugate gradients, in:Inverse and Ill-Posed Problems, eds. H.W. Engl and C.W. Groetsch (Academic Press, Boston, 1987).Google Scholar
12. [12]
T.F. Chan and P.C. Hansen, Some applications of the rank revealing QR factorization, SIAM J. Sci. Stat. Comp. 13 (1992) 727–741.Google Scholar
13. [13]
J.A. Cochran,The Analysis of Linear Integral Equations (McGraw-Hill, New York, 1972).Google Scholar
14. [14]
D. Colton and R. Kress,Integral Equation Methods for Scattering Theory (Wiley, New York, 1983).Google Scholar
15. [15]
L.J.D. Craig and J.C. Brown,Inverse Problems in Astronomy (Adam Hilger, Bristol, 1986).Google Scholar
16. [16]
J.J.M. Cuppen, A numerical soluiton of the inverse problem of electrocardiology, Ph.D. Thesis, Dept. of Mathematics, Univ. of Amsterdam (1983).Google Scholar
17. [17]
L.M. Delves and J.L. Mohamed,Computational Methods for Integral Equations (Cambridge University Press, Cambridge, 1985).Google Scholar
18. [18]
L.M. Delves and J. Walsh (eds.),Numerical Soluution of Integral Equations (Clarendon Press, Oxford, 1974).Google Scholar
19. [19]
J.B. Drake, ARIES: a computer program for the solution of first kind integral equations with noisy data, Report K/CSD/TM43, Dept. of Computer Science, Oak Ridge National Laboratory (October 1983).Google Scholar
20. [20]
M.P. Ekstrom and R.L. Rhodes, On the application of eigenvector expansions to numerical deconvolution, J. Comp. Phys. 14 (1974) 319–340.Google Scholar
21. [21]
L. Eldén, A program for interactive regularization, Report LiTH-MAT-R-79-25, Dept. of Mathematics, Linköping University, Sweded (1979).Google Scholar
22. [22]
L. Eldén, Algorithms for regularization of ill-conditioned least-squares problems, BIT 17 (1977) 134–145.Google Scholar
23. [23]
L. Eldén, A weighted pseudoinverse, generalized singular values, and constrained least squares problems, BIT 22 (1982) 487–501.Google Scholar
24. [24]
L. Eldén, A note on the computation of the genralized cross-validation function for illconditioned least squares problems, BIT 24 (1984) 467–472.Google Scholar
25. [25]
H.W. Engl and J. Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems, Appl. Numer. Math. 4 (1988) 395–417.Google Scholar
26. [26]
R.D. Fierro and J.R. Bunch, Collinearity and total least squares, Report, Dept. of Mathematics, Univ. of California, San Diego (1991), to appear in SIAM J. Matrix Anal. Appl.Google Scholar
27. [27]
R.D. Fierro, P.C. Hansen and D.P. O'Leary, Regularization and total least squares, manuscript in preparation for SIAM J. Sci. Comp.Google Scholar
28. [28]
R. Fletcher,Practical Optimization Methods. vol. 1, 2nd ed. (Wiley, Chichester, 1987).Google Scholar
29. [29]
G.H. Golub and C.F. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins, Baltimore, 1989).Google Scholar
30. [30]
G.H. Golub and U. von Matt, Quadratically constrained least squares and quadratic problems, Numer. Mathematik 59 (1991) 561–580.Google Scholar
31. [31]
C.W. Groetsch,The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).Google Scholar
32. [32]
C.W. Groetsch,Inverse Problems in the Mathematical Sciences (Vieweg Verlag, Wiesbaden), to be published.Google Scholar
33. [33]
C.W. Groetsch and C.R. Vogel, Asymptotic theory of filtering for linear operator equations with discrete noisy data, Math. Comp. 49 (1987) 499–506.Google Scholar
34. [34]
J. Hadamard,Lectures on Cauchy's Problem in Linear Differential Equations (Yale University Press, New Haven, 1923).Google Scholar
35. [35]
M. Hanke, Regularization with differential operators. An interative approach, J. Numer. Funct. Anal. Optim. 13 (1992) 523–540.Google Scholar
36. [36]
M. Hanke, Iterative solution of undertermined linear systems by transformation to standard form, submitted to J. Num. Lin. Alg. Appl.Google Scholar
37. [37]
M. Hanke and P.C. Hansen, Regularization methods for large-scale problems, Report UNIC-92-04, UNI.C (August 1992), to appear in Surveys Math. Ind.Google Scholar
38. [38]
P.C. Hansen, The truncated SVD as a method for regularization, BIT 27 (1987) 543–553.Google Scholar
39. [39]
P.C. Hansen, Computation of the singular value expansion, Computing 40 (1988) 185–199.Google Scholar
40. [40]
P.C. Hansen, Perturbation bounds for discrete Tikhonov regularization, Inverse Problems 5 (1989) L41-L44.Google Scholar
41. [41]
P.C. Hansen, Regularization, GSVD and truncated GSVD, BIT 29 (1989) 491–504.Google Scholar
42. [42]
P.C. Hansen, Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Statist. Comp. 11 (1990) 503–518.Google Scholar
43. [43]
P.C. Hansen, Relations between SVD and GSVD of discrete regularization problems in standard and general form, Lin. Alg. Appl. 141 (1990) 165–176.Google Scholar
44. [44]
P.C. Hansen, The discrete Picard condition for discrete ill-posed problems, BIT 30 (1990) 658–672.Google Scholar
45. [45]
P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580.Google Scholar
46. [46]
P.C. Hansen, Numerical tools for analysis and solution of Fredhold integral equations of the first kind, Inverse Problems 8 (1992) 849–872.Google Scholar
47. [47]
P.C. Hansen, Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems; Version 2.0 for Matlab 4.0, Report UNIC-92-03 (March 1993). Available in PostScript form by e-mail via netlib (netlib@research.att.com) from the library NUMERALGO.Google Scholar
48. [48]
P.C. Hansen and D.P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, Report CS-TR-2781, Dept. of Computer Science, Univ. of Maryland (1991), to appear in SIAM J. Sci. Comp.Google Scholar
49. [49]
P.C. Hansen, D.P. O'Leary and G.W. Stewart, Regularizing properties of conjugate gradient iterations, work in progress.Google Scholar
50. [50]
P.C. Hansen, T. Sekii and H. Shibahashi, The modified truncate SVD method for regularization is general form, SIAM J. Sci. Statist. Comp. 13 (1992) 1142–1150.Google Scholar
51. [51]
B. Hofmann,Regularization for Applied Inverse and Ill-Posed Problems (Teubner, Leipzig, 1986).Google Scholar
52. [52]
T. Kitagawa, A deterministic approach to optimal regularization — the finite dimensional case, Japan J. Appl. Math. 4 (1987) 371–391.Google Scholar
53. [53]
R. Kress,Linear Integral Equations (Springer, Berlin, 1989).Google Scholar
54. [54]
C.L. Lawson and R.J. Hanson,Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, 1974).Google Scholar
55. [55]
P. Linz, Uncertainty in the solution of linear operator equations, BIT 24 (1984) 92–101.Google Scholar
56. [56]
Matlab Reference Guide (The Math Works, MA, 1992).Google Scholar
57. [57]
K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1 (1970) 52–74.Google Scholar
58. [58]
V.A. Morozov,Methods for Solving Incorrectly Posed Problems (Springer, New York, 1984).Google Scholar
59. [59]
F. Natterer,The Mathematics of Computerized Tomography (Wiley, New York, 1986).Google Scholar
60. [60]
F. Natterer, Numerical treatment of ill-posed problems, in —.Google Scholar
61. [61]
D.P. O'Leary and J.A. Simmons, A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Stat. Comp. 2 (1981) 474–489.Google Scholar
62. [62]
C.C. Paige and M.A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8 (1982) 43–71.Google Scholar
63. [63]
R.L. Parker, Understanding inverse theory, Ann. Rev. Earth Planet Sci. 5 (1977), 35–64.Google Scholar
64. [64]
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
65. [65]
F. Santosa, Y.-H. Pao, W.W. Symes and C. Holland (eds.),Inverse Problems of Acoustic and Elastic Waves (SIAM, Philadelphia, 1984).Google Scholar
66. [66]
C. Ray Smith and W.T. Grandy, Jr. (eds.),Maximum-Entropy and Bayesian Methods in Inverse Problems (Reidel, Boston, 1985).Google Scholar
67. [67]
G. Talenti,Inverse Problems, Lecture Notes in Mathematics, 1225 (Springer, Berlin, 1986).Google Scholar
68. [68]
H.J.J. te Tiele, A program for solving first kind Fredholm integral equations by means of regularization, Comp. Phys. Comm. 36 (1985) 423–432.Google Scholar
69. [69]
A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Dokl. Adak. Nauk. SSSR 151 (1963) 501–504 [Soviet Math. Dokl. 4 (1963) 1035–1038].Google Scholar
70. [70]
A.N. Tikhonov and V.Y. Arsenin,Solutions of Ill-Posed Problems (Winston, Washington, DC, 1977).Google Scholar
71. [71]
A.N. Tikhonov and A.V. Goncharsky,Ill-Posed Problems in the Natural Sciences (MIR, Moscow, 1987).Google Scholar
72. [72]
A. van der Sluis, The convergence behavior of conjugate gradients and Ritz values in various circumstances, in:Iterative Methods in Linear Algebra, eds. R. Bauwens and P. de Groen (North-Holland, Amsterdam, 1992).Google Scholar
73. [73]
A. van der Sluis and H.A. van der Vorst, SIRT- and CG-type methods for iterative solution of sparse linear least-squares problems, Lin. Alg. Appl. 130 (1990) 257–302.Google Scholar
74. [74]
J.M. Varah, On the numerical solution of ill-conditioned linear systems with applications to illposed problems, SIAM J. Numer. Anal. 10 (1973) 257–267.Google Scholar
75. [75]
J.M. Varah, A practical examination of some numerical methods for linear discrete ill-posed problems, SIAM Rev. 21 (1979) 100–111.Google Scholar
76. [76]
J.M. Varah, Pitfalls in the numerical solution of linear ill-possed problems, SIAM J. Sci. Statist. Comp. 4 (1983) 164–176.Google Scholar
77. [77]
C.R. Vogel, Optimal choice of a truncationlevel for the standard SVD solutio of linear first kind integral equations when data are noisy, SIAM J. Numer. Anal. 23 (1986) 109–117.Google Scholar
78. [78]
C.R. Vogel, Solving ill-conditioned linear systems using the conjugate gradient method, Report, Dept. of Mathematical Sciences, Montana State University (1987).Google Scholar
79. [79]
G. Wahba,Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59 (SIAM, Philidelphia, 1990).Google Scholar
80. [80]
G.M. Wing, Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind, J. Integral Equations 9 (Suppl.) (1985) 191–204.Google Scholar
81. [81]
G.M. Wing and J.D. Zahrt,A Primer on Integral Equations of the First Kind (SIAM, Philidelphia, 1991).Google Scholar
82. [82]
H. Zha and P.C. Hansen, Regularization and the general Gauss-Markov linear model, Math. Comp. 55 (1990) 613–624.Google Scholar