October  2021, 15(5): 1035-1050. doi: 10.3934/ipi.2021027

On Tikhonov-type regularization with approximated penalty terms

Industrial Mathematics Institute, Johannes Kepler University, A-4040 Linz, Austria

Received  November 2020 Revised  January 2021 Published  October 2021 Early access  March 2021

In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term $ P_0 $, we approximate it by a family of penalty terms $ ({P_\beta}) $ having nicer properties and analyze what happens as $ \beta\to 0 $.

We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.

Citation: Andreas Neubauer. On Tikhonov-type regularization with approximated penalty terms. Inverse Problems and Imaging, 2021, 15 (5) : 1035-1050. doi: 10.3934/ipi.2021027
References:
[1]

A. B. Bakushinskii and A. Goncharsky, Iterative Methods for the Solution of Incorrect Problems, Nauka, Moscow, 1989, In Russian.

[2]

A. B. Bakushinskii and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, vol. 577 of Mathematics and Its Applications, Springer, Dordrecht, 2004.

[3]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

[4]

S. F. Gilyazov and N. L. Gol'dman, Regularization of Ill-Posed Problems by Iteration Methods, Kluwer, Dordrecht, 2000.

[5]

M. Hanke, A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.

[6]

P. C. Hansen and J. S. Jørgensen, AIR tools II: Algebraic iterative reconstruction methods, improved implementation, Numer. Algorithms, 79 (2018), 107-137.  doi: 10.1007/s11075-017-0430-x.

[7]

F. Hinterer, S. Hubmer and R. Ramlau, A note on the minimization of a Tikhonov functional with $\ell^1$-penalty, Inverse Problems, 36 (2020), 074001 19pp. doi: 10.1088/1361-6420/ab89c2.

[8]

K. Ito and B. Jin, Inverse Problems: Tikhonov Theory and Algorithms, vol. 22 of Series on Applied Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

[9]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, no. 6 in Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[10]

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.  doi: 10.1145/355984.355989.

[11]

C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Problems, 26 (2010), 105017. doi: 10.1088/0266-5611/26/10/105017.

[12]

R. Ramlau and C. A. Zarzer, On the minimization of a Tikhonov functional with a non-convex sparsity constraint, Electron. Trans. Numer. Anal., 39 (2012), 467-505. 

[13]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, vol. 167 of Applied Mathematical Sciences, Springer, New York, 2009.

[14]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods In Banach Spaces, vol. 10 of Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, 2012. doi: 10.1515/9783110255720.

[15]

A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, no. 14 in Vol. 1 and 2, Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1998. doi: 10.1007/978-94-017-5167-4.

show all references

References:
[1]

A. B. Bakushinskii and A. Goncharsky, Iterative Methods for the Solution of Incorrect Problems, Nauka, Moscow, 1989, In Russian.

[2]

A. B. Bakushinskii and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, vol. 577 of Mathematics and Its Applications, Springer, Dordrecht, 2004.

[3]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

[4]

S. F. Gilyazov and N. L. Gol'dman, Regularization of Ill-Posed Problems by Iteration Methods, Kluwer, Dordrecht, 2000.

[5]

M. Hanke, A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.

[6]

P. C. Hansen and J. S. Jørgensen, AIR tools II: Algebraic iterative reconstruction methods, improved implementation, Numer. Algorithms, 79 (2018), 107-137.  doi: 10.1007/s11075-017-0430-x.

[7]

F. Hinterer, S. Hubmer and R. Ramlau, A note on the minimization of a Tikhonov functional with $\ell^1$-penalty, Inverse Problems, 36 (2020), 074001 19pp. doi: 10.1088/1361-6420/ab89c2.

[8]

K. Ito and B. Jin, Inverse Problems: Tikhonov Theory and Algorithms, vol. 22 of Series on Applied Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

[9]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, no. 6 in Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. doi: 10.1515/9783110208276.

[10]

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.  doi: 10.1145/355984.355989.

[11]

C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Problems, 26 (2010), 105017. doi: 10.1088/0266-5611/26/10/105017.

[12]

R. Ramlau and C. A. Zarzer, On the minimization of a Tikhonov functional with a non-convex sparsity constraint, Electron. Trans. Numer. Anal., 39 (2012), 467-505. 

[13]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, vol. 167 of Applied Mathematical Sciences, Springer, New York, 2009.

[14]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods In Banach Spaces, vol. 10 of Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, 2012. doi: 10.1515/9783110255720.

[15]

A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, no. 14 in Vol. 1 and 2, Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1998. doi: 10.1007/978-94-017-5167-4.

Figure 1.  Exact solution and reconstructions for noise level $ \delta = 1 $%
Figure 2.  Exact solution and reconstructions for noise level $ \delta = 10 $%
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