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

Daniel Gerth; Bernd Hofmann; Daniel Gerth; Bernd Hofmann

doi:10.3934/math.2019.4.1223

AIMS Mathematics

2019, Volume 4, Issue 4: 1223-1247. doi: 10.3934/math.2019.4.1223

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Research article Topical Sections

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

Daniel Gerth ^,,
Bernd Hofmann

Faculty for Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany

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.
- regularization,
- inverse problems,
- linear ill-posed operator equations,
- sparsity,
- $\ell^1$-regularization,
- Tikhonov functional,
- oversmoothing penalty,
- convergence rate
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

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Abstract

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.

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