A modified Tikhonov regularization method. (English) Zbl 1328.65096
Summary: Tikhonov regularization and truncated singular value decomposition (TSVD) are two elementary techniques for solving a least squares problem from a linear discrete ill-posed problem. Based on these two techniques, a modified regularization method is proposed, which is applied to an Arnoldi-based hybrid method. Theoretical analysis and numerical examples are presented to illustrate the effectiveness of the method.
MSC:
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
References:
| [1] | Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer: Kluwer Dordrecht · Zbl 0859.65054 |
| [2] | Hansen, P. C., Rank-Deficient and Discrete Ill-Posed Problems (1998), SIAM: SIAM Philadelphia |
| [3] | Hansen, P. C., Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Stat. Comput., 11, 503-518 (1990) · Zbl 0699.65029 |
| [4] | Brezinski, C.; Redivo-Zaglia, M.; Rodriguez, G.; Seatzu, S., Multi-parameter regularization techniques for ill-conditioned linear systems, Numer. Math., 94, 203-228 (2003) · Zbl 1024.65036 |
| [5] | Calvetti, D.; Reichel, L.; Shuibi, A., Invertible smoothing preconditioners for linear discrete ill-posed problems, Appl. Numer. Math., 54, 135-149 (2005) · Zbl 1072.65057 |
| [6] | Fuhry, M.; Reichel, L., A new Tikhonov regularization method, Numer. Algorithms, 59, 433-445 (2012) · Zbl 1236.65038 |
| [7] | Reichel, L.; Ye, Q., Simple square smoothing regularization operators, Electron. Trans. Numer. Anal., 33, 63-83 (2009) · Zbl 1171.65033 |
| [8] | Brezinski, C.; Rodriguez, G.; Seatzu, S., Error estimates for linear systems with application to regularization, Numer. Algorithms, 49, 85-104 (2008) · Zbl 1162.65018 |
| [9] | Kindermann, S., Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems, Electron. Trans. Numer. Anal., 38, 233-257 (2011) · Zbl 1287.65043 |
| [10] | Reichel, L.; Shyshkov, A., A new zero-finder for Tikhonov regularization, Numer. Math., 48, 627-643 (2008) · Zbl 1161.65029 |
| [11] | Baglama, J.; Reichel, L., Decomposition methods for large linear discrete ill-posed problems, J. Comput. Appl. Math., 198, 332-343 (2007) · Zbl 1106.65035 |
| [12] | Vogel, C. R., Computational Methods for Inverse Problems (2002), SIAM · Zbl 1008.65103 |
| [13] | Morigi, S.; Reichel, L.; Sgallari, F., Iterative methods for ill-posed problems and semiconvergent sequences, J. Comput. Appl. Math., 193, 157-167 (2006) · Zbl 1092.65025 |
| [14] | Hansen, P. C., Discrete Inverse Problems: Insight and Algorithms (2010), SIAM · Zbl 1197.65054 |
| [15] | Calvetti, D.; Lewis, B.; Reichel, L., On the regularizing properties of the GMRES method, Numer. Math., 91, 605-625 (2002) · Zbl 1022.65044 |
| [16] | Eldén, L.; Simoncini, V., Solving ill-posed linear systems with GMRES and a singular preconditioner, SIAM J. Matrix Anal. Appl., 33, 1369-1394 (2012) · Zbl 1263.65033 |
| [18] | Hanke, M., On Lanczos based methods for the regularization of discrete ill-posed problems, BIT, 41, 1008-1018 (2001) |
| [19] | Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [20] | Bunse-Gerstner, A.; Guerra-Ones, V.; de La Vega, H. M., An improved preconditioned LSQR for discrete ill-posed problems, Math. Comput. Simulation, 73, 65-75 (2006) · Zbl 1104.65034 |
| [21] | Hochstenbach, M. E.; Reichel, L., An iterative method for Tikhonov regularization with a general linear regularization operator, J. Integral Equations Appl., 22, 465-482 (2010) · Zbl 1210.65092 |
| [22] | Morigi, S.; Reichel, L.; Sgallari, F., A truncated projected SVD method for linear discrete ill-posed problems, Numer. Algorithms, 43, 197-213 (2006) · Zbl 1114.65039 |
| [23] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0865.65009 |
| [24] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), Sci. Industrial Appl. Math. · Zbl 1002.65042 |
| [25] | Phillips, D. L., A technique for the numerical solution of certain integral equations of the first kind, J. ACM, 9, 84-97 (1962) · Zbl 0108.29902 |
| [26] | Hansen, P. C., Regularization tools version 4.0 for MATLAB 7.3, Numer. Algorithms, 46, 189-194 (2007) · Zbl 1128.65029 |
| [27] | Wing, G. M., A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding (1991), SIAM · Zbl 0753.45001 |
| [28] | Shaw, C. B., Improvement of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl., 37, 83-112 (1972) · Zbl 0237.65086 |
| [29] | Varah, J. M., Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Stat. Comput., 4, 164-176 (1983) · Zbl 0533.65082 |
| [30] | Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press · Zbl 0592.65093 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
