A Data-Driven Iteratively Regularized Landweber Iteration

Abstract

We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.

Solving inverse problems using data-driven models

“…The (possibly non-linear) operator B : X → Y can now be represented by a deep neural network that can be trained against supervised data by comparing the final iterate in (6.6) (iterates are stopped following the Morozov discrepancy principle (4.1)) against the ground truth for given data. Convergence and stability for the scheme in (6.6) in infinite-dimensional Hilbert spaces is proved by Aspri et al (2018). This theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a non-linear inverse problem of Schlieren tomography.…”

“…Convergence and stability for the scheme in (6.6) in infinite-dimensional Hilbert spaces is proved by Aspri et al. (2018). This theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a non-linear inverse problem of Schlieren tomography.…”

“…Another approach to including a learning component into a forward operator is presented by Aspri, Banert, Öktem and Scherzer (2018). The starting point is to consider the iteratively regularized Landweber iteration (Scherzer 1998) (see also Kaltenbacher et al.…”

“…This theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a non-linear inverse problem of Schlieren tomography. In these examples, however, Aspri et al (2018) restrict attention to a linear operator B.…”

Data driven regularization by projection

“…Nowadays, with the rise of the area of big data, methods that combine forward modeling with data driven techniques are being developed [7]. Some of these techniques build upon the similarity between deep neural networks and classical approaches to inverse problems such as iterative regularization [8,9] and proximal methods [10]. Some are based on postprocessing of the reconstructions obtained by a simple inversion technique such as filtered backprojection [11].…”