Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators

Abstract

In this paper we deal with Morozov's discrepancy principle as an aposteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for non-linear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for this parameter choice rule holds α → 0, δ q /α → 0 as the noise level δ goes to 0. It is illustrated that for suitable penalty terms this yields convergence of the regularized solutions to the true solution in the topology induced by Ψ. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.

On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

“…By Proposition 3, if we also take the limit m → ∞, the resulting sequence has a weakly convergent subsequence with limit satisfying the continuous version of the Morozov discrepancy principle presented in [4]. For this reason, we can always assume the existence of a diagonal subsequence converging (weakly) to an f x 0 -minimizing solution of Problem 1 when δ → 0.…”

“…Thus, we base our choice of both parameters on the same relaxed version of Morozov's discrepancy principle. The present approach applies in a nontrivial way the methodology developed in [16,6,4,5] to the context of nonlinear operators under a discrete setting. We also establish that the continuous case can be recovered from the discrete one, when the discretization level goes to infinity.…”

“…Let m and α satisfy the discrepancy principle (12). Assume that for the same m and α, H m is nonempty and x δ m,α is a minimizer of (4). Assume that F is Frechét differentiable in x † and let the source condition…”

A modified coupled complex boundary method for an inverse chromatography problem

“…Next, we develop the following a posteriori parameter choice rule -the Morozov's discrepancy principle [1]:…”

An adaptive parameter estimation for guided filter based image deconvolution

“…It depends only on the data and automatically computes the regularization parameter according to the data. The Morozov's discrepancy principle [39] is useful for the selection of λ when the noise variance is available. Based on this principle, for an image of N × N size, a good estimation of the deconvolution problem should lie in set…”