A stochastic convergence analysis for Tikhonov regularization with sparsity constraints

Gerth, Daniel; Ramlau, Ronny

doi:10.1088/0266-5611/30/5/055009

Cited by 8 publications

(37 citation statements)

References 20 publications

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“…We now follow the construction in [22] to define a Besov prior using the wavelet basis. Notice also the work in [7,13] on non-linear problems and Besov prior on the full space R d , respectively.…”

Section: Example 1: Besov Priormentioning

confidence: 99%

Maximuma posterioriprobability estimates in infinite-dimensional Bayesian inverse problems

2015

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A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in highdimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Current inverse problem literature often approaches the problem by considering suitable point estimators for the task. Typically the choice is made between the maximum a posteriori (MAP) or the conditional mean (CM) estimate.The benefits of either choice are not well-understood from the perspective of infinite-dimensional theory. Most importantly, there exists no general scheme regarding how to connect the topological description of a MAP estimate to a variational problem. The recent results by Dashti and others [8] resolve this issue for non-linear inverse problems in Gaussian framework. In this work we improve the current understanding by introducing a novel concept called the weak MAP (wMAP) estimate. We show that any MAP estimate in the sense of [8] is a wMAP estimate and, moreover, how the wMAP estimate connects to a variational formulation in general infinite-dimensional non-Gaussian problems. The variational formulation enables to study many properties of the infinitedimensional MAP estimate that were earlier impossible to study.In a recent work by the authors [6] the MAP estimator was studied in the context of the Bayes cost method. Using Bregman distances, proper convex Bayes cost functions were introduced for which the MAP estimator is the Bayes estimator. Here, we generalize these results to the infinite-dimensional setting. Moreover, we discuss the implications of our results for some examples of prior models such as the Besov prior and hierarchical prior.

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Section: Example 1: Besov Priormentioning

confidence: 99%

Maximuma posterioriprobability estimates in infinite-dimensional Bayesian inverse problems

2015

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“…We also would like to mention convergence speed results, which require, e.g., source conditions for the solution [2,3,8,14,26,27,41,51]. Convergence and convergence rates for sparse regularization in a Bayes setup have been recently investigated in [25].…”

Section: Tikhonov Regularization With Sparsity Constraintsmentioning

confidence: 99%

Wavelet methods for a weighted sparsity penalty for region of interest tomography

2015

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We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓ ppenalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.

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“…for some kernel function k ∈ L 2 (R d ). If the Fourier transformk of k satisfies |k(ξ)| ∼ (1 + |ξ| 2 ) −β/2 , then (2.5) holds with l = β, see, e.g., [9]. ✷ Assume that the minimal norm solution x of (1.1) lives in H k .…”

Section: Discretization Of the Operator Equationmentioning

confidence: 99%

“…then (2.5) holds with l = β, see, e.g., [9]. ✷ Assume that the minimal norm solution x of (1.1) lives in H k .…”

Section: Discretization Of the Operator Equationmentioning

confidence: 99%

Error estimates for Arnoldi–Tikhonov regularization for ill-posed operator equations

Ramlau

Reichel

2019

Inverse Problems

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Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.

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A stochastic convergence analysis for Tikhonov regularization with sparsity constraints

Cited by 8 publications

References 20 publications

Maximuma posterioriprobability estimates in infinite-dimensional Bayesian inverse problems

Maximuma posterioriprobability estimates in infinite-dimensional Bayesian inverse problems

Wavelet methods for a weighted sparsity penalty for region of interest tomography

Error estimates for Arnoldi–Tikhonov regularization for ill-posed operator equations

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