May  2014, 8(2): 409-420. doi: 10.3934/ipi.2014.8.409

An inner-outer regularizing method for ill-posed problems

1. 

IIT - CNR Via G. Moruzzi 1, 56124 Pisa, Italy

2. 

Dipart. di Matematica e Informatica, University of Parma, Viale G. Usberti 53/A, 43100 Parma, Italy

3. 

Dipart. di Informatica, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy, Italy

Received  September 2013 Revised  February 2014 Published  May 2014

Conjugate Gradient is widely used as a regularizing technique for solving linear systems with ill-conditioned coefficient matrix and right-hand side vector perturbed by noise. It enjoys a good convergence rate and computes quickly an iterate, say $x_{k_{opt}}$, which minimizes the error with respect to the exact solution. This behavior can be a disadvantage in the regularization context, because also the high-frequency components of the noise enter quickly the computed solution, leading to a difficult detection of $k_{opt}$ and to a sharp increase of the error after the $k_{opt}$th iteration. In this paper we propose an inner-outer algorithm based on a sequence of restarted Conjugate Gradients, with the aim of overcoming this drawback. A numerical experimentation validates the effectiveness of the proposed algorithm.
Citation: Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems and Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409
References:
[1]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, 1998. doi: 10.1887/0750304359.

[2]

B. Eicke, A. K. Louis and R. Plato, The instability of some gradient methods for ill-posed problems, Numer. Math., 58 (1990), 129-134. doi: 10.1007/BF01385614.

[3]

P. Favati, G. Lotti, O. Menchi and F. Romani, Generalized Cross-Validation Applied to Conjugate Gradient for Discrete Ill-Posed Problems, Technical Report IIT TR-09/2013. Available from: http://www.iit.cnr.it/sites/default/files/TR-09-2013.pdf

[4]

D. A. Girard, A fast ‘Monte-Carlo Cross-Validation' procedure for large least squares problems with noisy data, Numer. Math., 56 (1989), 1-23. doi: 10.1007/BF01395775.

[5]

G. H. Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223. doi: 10.1080/00401706.1979.10489751.

[6]

M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics, Longman, Harlow, 1995.

[7]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM Monographs on Mathematical Modeling and Computation, Philadelphia, 1998. doi: 10.1137/1.9780898719697.

[8]

P. C. Hansen, Regularization tools, Numer. Algo., 46 (2007), 189-194. doi: 10.1007/s11075-007-9136-9.

[9]

B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle, IMA J. Numer. Anal., 32 (2012), 1714-1732. doi: 10.1093/imanum/drr051.

[10]

K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Problems, 14 (1998), 1247-1264. doi: 10.1088/0266-5611/14/5/010.

[11]

R. Plato, Optimal algorithms for linear ill-posed problems yield regularization methods, Num. Funct. Anal. and Optimiz., 11 (1990), 111-118. doi: 10.1080/01630569008816364.

[12]

R. J. Santos and A. R. De Pierro, The effect of the nonlinearity on GCV applied to Conjugate Gradients in computerized tomography, Comput. Appl. Math., 25 (2006), 111-128. doi: 10.1590/S0101-82052006000100006.

[13]

C. Vogel, Computational Methods for Inverse Problems, SIAM Frontier in Applied Mathematics, Philadelphia, 2002. doi: 10.1137/1.9780898717570.

[14]

G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., 14 (1977), 651-667. doi: 10.1137/0714044.

[15]

Y. Wang, A restarted conjugate gradient method for ill-posed problems, Acta Mathematicae Applicatae Sinica, English Series, 19 (2003), 31-40. doi: 10.1007/s10255-003-0078-2.

[16]

J. Xie and J. Zou, An improved model function method for choosing regularization parameters in linear inverse problems, Inverse Problems, 18 (2002), 631-643. doi: 10.1088/0266-5611/18/3/307.

[17]

F. Zama and E. Loli Piccolomini, A descent method for regularization of ill-posed problems, Optimization Methods and Software, 20 (2005), 615-628. doi: 10.1080/10556780500140409.

show all references

References:
[1]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, 1998. doi: 10.1887/0750304359.

[2]

B. Eicke, A. K. Louis and R. Plato, The instability of some gradient methods for ill-posed problems, Numer. Math., 58 (1990), 129-134. doi: 10.1007/BF01385614.

[3]

P. Favati, G. Lotti, O. Menchi and F. Romani, Generalized Cross-Validation Applied to Conjugate Gradient for Discrete Ill-Posed Problems, Technical Report IIT TR-09/2013. Available from: http://www.iit.cnr.it/sites/default/files/TR-09-2013.pdf

[4]

D. A. Girard, A fast ‘Monte-Carlo Cross-Validation' procedure for large least squares problems with noisy data, Numer. Math., 56 (1989), 1-23. doi: 10.1007/BF01395775.

[5]

G. H. Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223. doi: 10.1080/00401706.1979.10489751.

[6]

M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics, Longman, Harlow, 1995.

[7]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM Monographs on Mathematical Modeling and Computation, Philadelphia, 1998. doi: 10.1137/1.9780898719697.

[8]

P. C. Hansen, Regularization tools, Numer. Algo., 46 (2007), 189-194. doi: 10.1007/s11075-007-9136-9.

[9]

B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle, IMA J. Numer. Anal., 32 (2012), 1714-1732. doi: 10.1093/imanum/drr051.

[10]

K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Problems, 14 (1998), 1247-1264. doi: 10.1088/0266-5611/14/5/010.

[11]

R. Plato, Optimal algorithms for linear ill-posed problems yield regularization methods, Num. Funct. Anal. and Optimiz., 11 (1990), 111-118. doi: 10.1080/01630569008816364.

[12]

R. J. Santos and A. R. De Pierro, The effect of the nonlinearity on GCV applied to Conjugate Gradients in computerized tomography, Comput. Appl. Math., 25 (2006), 111-128. doi: 10.1590/S0101-82052006000100006.

[13]

C. Vogel, Computational Methods for Inverse Problems, SIAM Frontier in Applied Mathematics, Philadelphia, 2002. doi: 10.1137/1.9780898717570.

[14]

G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., 14 (1977), 651-667. doi: 10.1137/0714044.

[15]

Y. Wang, A restarted conjugate gradient method for ill-posed problems, Acta Mathematicae Applicatae Sinica, English Series, 19 (2003), 31-40. doi: 10.1007/s10255-003-0078-2.

[16]

J. Xie and J. Zou, An improved model function method for choosing regularization parameters in linear inverse problems, Inverse Problems, 18 (2002), 631-643. doi: 10.1088/0266-5611/18/3/307.

[17]

F. Zama and E. Loli Piccolomini, A descent method for regularization of ill-posed problems, Optimization Methods and Software, 20 (2005), 615-628. doi: 10.1080/10556780500140409.

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