On the lifting of deterministic convergence rates for inverse problems with stochastic noise

Daniel Gerth; Andreas Hofinger; Ronny Ramlau

doi:10.3934/ipi.2017031

Article Contents

2017, Volume 11, Issue 4: 663-687. Doi: 10.3934/ipi.2017031

This issue Previous Article Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery Next Article Non-convex TV denoising corrupted by impulse noise

On the lifting of deterministic convergence rates for inverse problems with stochastic noise

1.
Technische Universität Chemnitz, Fakultät für Mathematik, D-09107 Chemnitz, Germany
2.
Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstraße 69, A-4040 Linz, Austria
3.
Radon Institute for Computational and Applied Mathematics (RICAM), (also Industrial Mathematics Institute, Johannes Kepler University Linz), Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Daniel Gerth
* Corresponding author: Daniel Gerth

Received: April 2016

Revised: April 2017

Published: October 2017

The first author was supported in part by the Austrian Science Fund (FWF): W1214-N15 and by the German Research Foundation (DFG) under grants HO1454/8-2 and HO1454/10-1.

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Abstract

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is crucial. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.

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Mathematics Subject Classification: Primary: 60H35, 65R32; Secondary: 47J06.

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Figure 1. $\mathbb{E}(||\epsilon||)^2/ \alpha$ (dashed) and regularization parameter $\alpha$ (solid) versus variance $\eta$. Left: A constant value of $\tau$ in the discrepancy principle with the expectation of the noise leads to the regularization parameter decreasing too fast, thus the deterministic condition $\delta^2/\alpha\rightarrow0$ is violated (dashed line) Right: increasing $\tau$ appropriately with decreasing variance resolves this issue

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