# American Institute of Mathematical Sciences

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 & 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. Google Scholar [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.  Google Scholar [3] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.  Google Scholar [4] S. F. Gilyazov and N. L. Gol'dman, Regularization of Ill-Posed Problems by Iteration Methods, Kluwer, Dordrecht, 2000. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] A. B. Bakushinskii and A. Goncharsky, Iterative Methods for the Solution of Incorrect Problems, Nauka, Moscow, 1989, In Russian. Google Scholar [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.  Google Scholar [3] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.  Google Scholar [4] S. F. Gilyazov and N. L. Gol'dman, Regularization of Ill-Posed Problems by Iteration Methods, Kluwer, Dordrecht, 2000. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Exact solution and reconstructions for noise level $\delta = 1$%
Exact solution and reconstructions for noise level $\delta = 10$%
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