### AIMS Mathematics

2019, Issue 4: 1223-1247. doi: 10.3934/math.2019.4.1223
Research article Topical Sections

# Oversmoothing regularization with $\ell^1$-penalty term

• Received: 31 January 2019 Accepted: 13 August 2019 Published: 29 August 2019
• MSC : 47A52, 65J20

• In Tikhonov-type regularization for ill-posed problems with noisy data, the penalty functional is typically interpreted to carry a-priori information about the unknown true solution. We consider in this paper the case that the corresponding a-priori information is too strong such that the penalty functional is oversmoothing, which means that its value is infinite for the true solution. In the case of oversmoothing penalties, convergence and convergence rate assertions for the regularized solutions are difficult to derive, only for the Hilbert scale setting convincing results have been published. We attempt to extend this setting to $\ell^1$-regularization when the solutions are only in $\ell^2$. Unfortunately, we have to restrict our studies to the case of bounded linear operators with diagonal structure, mapping between $\ell^2$ and a separable Hilbert space. But for this subcase, we are able to formulate and to prove a convergence theorem, which we support with numerical examples.

Citation: Daniel Gerth, Bernd Hofmann. Oversmoothing regularization with $\ell^1$-penalty term[J]. AIMS Mathematics, 2019, 4(4): 1223-1247. doi: 10.3934/math.2019.4.1223

### Related Papers:

• In Tikhonov-type regularization for ill-posed problems with noisy data, the penalty functional is typically interpreted to carry a-priori information about the unknown true solution. We consider in this paper the case that the corresponding a-priori information is too strong such that the penalty functional is oversmoothing, which means that its value is infinite for the true solution. In the case of oversmoothing penalties, convergence and convergence rate assertions for the regularized solutions are difficult to derive, only for the Hilbert scale setting convincing results have been published. We attempt to extend this setting to $\ell^1$-regularization when the solutions are only in $\ell^2$. Unfortunately, we have to restrict our studies to the case of bounded linear operators with diagonal structure, mapping between $\ell^2$ and a separable Hilbert space. But for this subcase, we are able to formulate and to prove a convergence theorem, which we support with numerical examples.

 [1] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Dordrecht: Kluwer Academic Publishers, 1996. [2] A. K. Louis, Inverse und schlecht gestellt Probleme, Stuttgart: Teubner, 1989. [3] D. Gerth and B. Hofmann, On $\ell^1$-regularization under continuity of the forward operator in weaker topologies. In: New Trends in Parameter Identification for Mathematical Models (Ed.: B. Hofmann, A. Leitao, J.P. Zubelli), Cham: Birkhäuser, (2018), 67-88. [4] B. Hofmann, B. Kaltenbacher, C. Pöschl, et al. A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Probl., 23 (2007), 987-1010. doi: 10.1088/0266-5611/23/3/009 [5] B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Probl., 28 (2012), 104006. [6] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18 (1984), 29-37. doi: 10.1080/00036818408839508 [7] B. Hofmann and P. Mathé, Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales, Inverse Probl., 34 (2018), 015007. [8] B. Hofmann and P. Mathé, A priori parameter choice in Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems, 2019. Available from: https://arxiv.org/abs/1904.02014. [9] T. Schuster, B. Kaltenbacher, B. Hofmann, et al. Regularization Methods in Banach spaces, Berlin/Boston: De Gruyter, 2012. [10] B. Hofmann, On smoothness concepts in regularization for nonlinear inverse problems in Banach spaces. In: Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts (Ed.: R. Melnik), New Jersey: John Wiley, (2015), 192-221. [11] M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Probl., 29 (2013), 025013. [12] J. Flemming, B. Hofmann and I. Veselic, A unified approach to convergence rates for $\ell^1$-regularization and lacking sparsity, J. Inverse Ill-posed Probl., 24 (2016), 139-148. [13] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pur. Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042 [14] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term, Inverse Probl., 24 (2008), 055020. [15] D. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints, J. Inverse ill-posed Probl, 16 (2008), 463-478. [16] R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints, Electron. T. Numer. Anal., 30 (2008), 54-74. [17] J. Flemming, Convergence rates for l1-regularization without injectivity-type assumptions, Inverse Probl., 32 (2016), 095001. [18] D. Gerth, J. Flemming, Injectivity and weak*-to-weak continuity suffice for convergence rates in l1-regularization, J. Inverse Ill-posed Probl., 26 (2018), 85-94. doi: 10.1515/jiip-2017-0008 [19] D. Gerth, Convergence rates for $\ell^1$-regularization without the help of a variational inequality, Electron. T. Numer. Anal., 46 (2017), 233-244. [20] B. Hofmann, P. Mathé and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, J. Inverse Ill-posed Probl., 16 (2008), 567-585. [21] D. Gerth, Using Landweber iteration to quantify source conditions - a numerical study, J. Inverse Ill-posed Probl., 27 (2019), 367-383. doi: 10.1515/jiip-2018-0071
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.427 1.6

Article outline

## Figures and Tables

Figures(6)  /  Tables(3)

• On This Site