Convergence rates for an iteratively regularized Newton–Landweber iteration in Banach space

Abstract

In this paper, we provide convergence and convergence rate results for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting. Numerical experiments illustrate the performance of the method.

Accelerated iterative regularization via dual diagonal descent

“…Iterative regularization methods have been studied also for convex regularizers in [29] where estimates in terms of Bregman distance were proved (see, e.g., [30,18] for Tikhonov-type approaches), but no explicit rates in the form (1.2) were shown. More general iterative algorithms defined in Banach spaces have been studied in [42,43,28] for linear and non-linear inverse problems and in [26] for L 1 and Total Variation (TV) regularization. For results on iterative regularization for data-fit terms other than squared norm, we mention [24] for results in the framework of Bregman distances and [39] where a dual diagonal descent (3D) algorithm is considered.…”

Iterative Solution Methods

“…Results on Gauss-Newton methods with other regularization than Tikhonov (similarly to section "Generalizations of the IRGNM") can be found, e.g., in [58,68].…”

Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space

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This paper is a close follow-up of [9] and [10], where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases.

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“…We will here especially concentrate on a combination of a Newton-type strategy with Landweber iterations to approximate the Newton step, which leads to a fully explicit iteration, cf. [9,10].…”

“…It is clear that the choice of the parameters α n,k , ω n,k , k n , and of the overall stopping index n = n * crucially influences the stability and efficiency of the method. While in [9], [10] only either convergence or convergence rates have been established with disjoint parameter choices for each of these two cases, the aim of this paper is to provide a unified parameter choice strategy for this method that allows to show both unconditional convergence and convergence rates under additional regularity assumptions on the solution. Moreover, by using an appropriate choice of the stopping index for the inner iteration, differently from [9], [10] we can show continuous dependence of the iterates x n on the data y δ for each fixed outer iteration index n and therewith are able to prove strong convergence.…”