Journal of Scientific Computing

, Volume 63, Issue 1, pp 163–184

Simple and Efficient Determination of the Tikhonov Regularization Parameter Chosen by the Generalized Discrepancy Principle for Discrete Ill-Posed Problems

Article

Abstract

Discrete ill-posed problems where both the coefficient matrix and the right hand side are contaminated by noise appear in a variety of engineering applications. In this paper we consider Tikhonov regularized solutions where the regularization parameter is chosen by the generalized discrepancy principle (GDP). In contrast to Newton-based methods often used to compute such parameter, we propose a new algorithm referred to as GDP-FP, where derivatives are not required and where the regularization parameter is calculated efficiently by a fixed-point iteration procedure. The algorithm is globally and monotonically convergent. Additionally, a specialized version of GDP-FP based on a projection method, that is well-suited for large-scale Tikhonov problems, is also proposed and analyzed in detail. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms on test problems from the literature.

Keywords

Discrete ill-posed problems Tikhonov regularization  Projection method Generalized discrepancy principle Noisy operator Noisy right hand side

References

1. 1.
Ang, D.D., Gorenflo, R., Le, V.K., Trong, D.D.: Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction, Lect. Notes Math. 1792. Springer, Berlin (2002)Google Scholar
2. 2.
Bazán, F.S.V.: Fixed-point iterations in determining the Tikhonov regularization parameter. Inverse Probl. 24, 1–15 (2008)
3. 3.
Bazán, F.S.V., Francisco, J.B., Leem, K.H., Pelekanos, G.: A maximum product criterion as a Tikhonov parameter choice rule for Kirsch’s factorization method. J. Comput. Appl. Math. 236, 4264–4275 (2012)
4. 4.
Bazán, F.S.V., Borges, L.S.: GKB-FP:an algorithm for large-scale discrete ill-posed problems. BIT 50(3), 481–507 (2010)
5. 5.
Bazán, F.S.V., Cunha, M.C., Borges, L.S.: Extension of GKB-FP to large-scale general-form Tikhonov regularization. Numer. Linear Algebra Appl. 21, 316–339 (2014)
6. 6.
Björck, Å.: A bidiagonalization algorithm for solving ill-posed systems of linear equations. BIT 28, 659–670 (1988)
7. 7.
Chung, J., Nagy, J.G., O’Leary, D.P.: A weighted-GCV method for Lanczos-hybrid regularization. Electron. Trans. Numer. Anal. 28, 149–167 (2008)
8. 8.
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, New York (1992)
9. 9.
Colton, D., Piana, M., Potthast, R.: A simple method using Morozov’s discrepancy principle for solving inverse scattering problems. Inverse Probl. 13, 1477–1493 (1999)
10. 10.
Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least square problems. BIT 22, 487–502 (1982)
11. 11.
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems (Mathematics and its Applications), vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)
12. 12.
Golub, G.H., Hansen, P.C., O’Leary, D.P.: Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21, 185–194 (1999)
13. 13.
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
14. 14.
Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discreet Ill-posed problems. Numer. Algorithms 6, 1–35 (1994)
15. 15.
Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)
16. 16.
Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22, 463–480 (2010)
17. 17.
Kirsch, A.: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Probl. 14, 1489–1512 (1998)
18. 18.
Lampe, J., Voss, H.: Large-scale Tikhonov regularization of total least squares. J. Comput. Appl. Math. 238, 95–108 (2013)
19. 19.
Leem, K.H., Pelekanos, G., Bazán, F.S.V.: Fixed-point iterations in determining a Tikhonov regularization parameter in Kirsch’s factorization method. Appl. Math. Comput. 216, 3747–3753 (2010)
20. 20.
Lu, S., Pereverzev, S.V., Tautenhahn, U.: Regularized total least squares: computational aspects and error bounds. SIAM J. Matrix Anal. Appl. 31, 918–941 (2009)
21. 21.
Lu, S., Pereverzev, S.V., Shao, Y., Tautenhahn, U.: On the generalized discrepancy principle for Tikhonov regularization in Hilbert Scales. J. Integral Equ. Appl. 22(3), 483–517 (2010)
22. 22.
Markovsky, I., Van Huffel, S.: Overview of total least squares methods. Signal Process. 87, 2283–2302 (2007)
23. 23.
Morozov, V.A.: Regularization Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)
24. 24.
Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8(1), 43–71 (1982)
25. 25.
Paige, C.C., Saunders, M.A.: Algorithm 583. LSQR: sparse linear equations and least squares problems. ACM Trans. Math. Softw. 8(2), 195–209 (1982)
26. 26.
Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63–83 (2009)
27. 27.
Renaut, R.A., Guo, G.: Efficient algorithms for solution of regularized total least squares. SIAM J. Matrix Anal. Appl. 21, 457–476 (2005)
28. 28.
Sima, D., Van Huffel, S., Golub, G.H.: Regularized total least squares based on quadratic eigenvalue problems solvers. BIT 44, 793–812 (2004)
29. 29.
Tautenhahn, U.: Regularization of linear ill-posed problems with noisy right hand side and noisy operator. J. Inverse Ill Posed Probl. 16, 507–523 (2008)
30. 30.
Tikhonov, A.N., Arsenin, V.Y.: Solution of Ill-Posed Problems. Wiley, New York (1977)Google Scholar
31. 31.
Vinokurov, V.A.: Two notes on the choice of regularization parameter. USSR Comput. Math. Math. Phys. 12(2), 249–253 (1972)