Anne Wald

Anne Wald, Thomas Schuster

In this work we discuss a method to adapt sequential subspace optimization (SESOP), which has so far been developed for linear inverse problems in Hilbert and Banach spaces, to the case of nonlinear inverse problems. We start by revising the well-known technique for Hilbert spaces. In a next step, we introduce a method using multiple search directions that are especially designed to fit the nonlinearity of the forward operator. To this end, we iteratively project the initial value onto stripes whose shape is determined by the search direction, the nonlinearity of the operator and the noise level. We additionally propose a fast algorithm that uses two search directions. Finally we will show convergence and regularization properties for the presented method.

Anne Wald

We introduce and analyze a fast iterative method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. The key idea, in contrast to the standard Landweber method, is to use multiple search directions per iteration in combination with a regulation of the step width in order to reduce the total number of iterations. This method is suitable for both exact and noisy data. In the latter case, we obtain a regularization method. An algorithm with two search directions is used for the numerical identification of a parameter in an elliptic boundary value problem.

Gesa Sarnighausen, Anne Wald, Andreas Hauptmann

We study the concept of including the causality principle as regularizer into the solution of linear time-dependent inverse problems. This is achieved by combining transformer-based predictions with classical variational regularization, resulting in what we call transformer causality regularization (TCR). The causality principle states that an object at time $t'$ depends only on its previous states at $t < t'$ and is independent of future states at $t > t'$. Since the transformer architecture represents sequence-to-sequence functions and can be equipped with a causal attention mask, transformers are the natural choice for a learned causality function that predicts the state of an object at time $t'$ given the previous states at $t < t'$. We combine this with the inductive bias of convolutional neural networks (CNNs) for imaging tasks to treat the spatial variable. The output of the spatial-temporal transformer is then used as a prior for variational regularization, such that classical results on regularization and convergence for solution methods directly transfer to our case. Using the example of dynamic computerized tomography, we compare TCR to a static and dynamic version of the earlier introduced unrolled adversarial regularizer for simulated and measured data. The results show that using TCR within a variational framework improves reconstruction results and data-consistency.