Sergiy Pereverzyev

Johannes Schwab, Sergiy Pereverzyev, Markus Haltmeier

The development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography. In this paper we address this issue for photoacoustic computed tomography in circular geometry. We investigate the Galerkin least squares method for that purpose. For approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single Funktion. This includes many systems that have previously been employed in PAT such as generalized Kaiser-Bessel basis functions or the natural pixel basis. By exploiting an isometry property of the forward problem we are able to efficiently set up the Galerkin equation for a wide class of generating functions and Devise efficient algorithms for its solution. We establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm.

Florian Dreier, Sergiy Pereverzyev, Markus Haltmeier

In photoacoustic tomography, one is interested to recover the initial pressure distribution inside a tissue from the corresponding measurements of the induced acoustic wave on the boundary of a region enclosing the tissue. In the limited view problem, the wave boundary measurements are given on the part of the boundary, whereas in the full view problem, the measurements are known on the whole boundary. For the full view problem, there exist various fast and robust reconstruction methods. These methods give severe reconstruction artifacts when they are applied directly to the limited view data. One approach for reducing such artefacts is trying to extend the limited view data to the whole region boundary, and then use existing reconstruction methods for the full view data. In this paper, we propose an operator learning approach for constructing an operator that gives an approximate extension of the limited view data. We consider the behavior of a reconstruction formula on the extended limited view data that is given by our proposed approach. Approximation errors of our approach are analyzed. We also present numerical results with the proposed extension approach supporting our theoretical analysis.

Stefan Kindermann, Sergiy Pereverzyev, Andrey Pilipenko

The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness of the linear functionals. We prove convergence rates for the linear functional strategy with these heuristic rules taking into account the smoothness of the solution and the functionals and imposing a structural condition on the noise. Furthermore, we study these noise conditions in both a deterministic and stochastic setup and verify that for mildly-ill-posed problems and Gaussian noise, these conditions are satisfied almost surely, where on the contrary, in the severely-ill-posed case and in a similar setup, the corresponding noise condition fails to hold. Moreover, we propose an aggregation method for adaptively optimizing the parameter choice rule by making use of improved rates for linear functionals. Numerical results indicate that this method yields better results than the standard heuristic rule.

Markus Holzleitner, Sergiy Pereverzyev, Sergei V. Pereverzyev et al.

This paper investigates a general regularization framework for unsupervised domain adaptation in vector-valued regression under the covariate shift assumption, utilizing vector-valued reproducing kernel Hilbert spaces (vRKHS). Covariate shift occurs when the input distributions of the training and test data differ, introducing significant challenges for reliable learning. By restricting the hypothesis space, we develop a practical operator learning algorithm capable of handling functional outputs. We establish optimal convergence rates for the proposed framework under a general source condition, providing a theoretical foundation for regularized learning in this setting. We also propose an aggregation-based approach that forms a linear combination of estimators corresponding to different regularization parameters and different kernels. The proposed approach addresses the challenge of selecting appropriate tuning parameters, which is crucial for constructing a good estimator, and we provide a theoretical justification for its effectiveness. Furthermore, we illustrate the proposed method on a real-world face image dataset, demonstrating robustness and effectiveness in mitigating distributional discrepancies under covariate shift.

Elke R. Gizewski, Markus Holzleitner, Lukas Mayer-Suess et al.

Most of the recent results in polynomial functional regression have been focused on an in-depth exploration of single-parameter regularization schemes. In contrast, in this study we go beyond that framework by introducing an algorithm for multiple parameter regularization and presenting a theoretically grounded method for dealing with the associated parameters. This method facilitates the aggregation of models with varying regularization parameters. The efficacy of the proposed approach is assessed through evaluations on both synthetic and some real-world medical data, revealing promising results.

Christoph Angermann, Markus Haltmeier, Ruth Steiger et al.

Convolutional neural networks are state-of-the-art for various segmentation tasks. While for 2D images these networks are also computationally efficient, 3D convolutions have huge storage requirements and require long training time. To overcome this issue, we introduce a network structure for volumetric data without 3D convolutional layers. The main idea is to include maximum intensity projections from different directions to transform the volumetric data to a sequence of images, where each image contains information of the full data. We then apply 2D convolutions to these projection images and lift them again to volumetric data using a trainable reconstruction algorithm.The proposed network architecture has less storage requirements than network structures using 3D convolutions. For a tested binary segmentation task, it even shows better performance than the 3D U-net and can be trained much faster.

Shuai Lu, Peter Mathé, Sergiy Pereverzyev

This paper studies a Nyström type subsampling approach to large kernel learning methods in the misspecified case, where the target function is not assumed to belong to the reproducing kernel Hilbert space generated by the underlying kernel. This case is less understood, in spite of its practical importance. To model such a case, the smoothness of target functions is described in terms of general source conditions. It is surprising that almost for the whole range of the source conditions, describing the misspecified case, the corresponding learning rate bounds can be achieved with just one value of the regularization parameter. This observation allows a formulation of mild conditions under which the plain Nyström subsampling can be realized with subquadratic cost maintaining the guaranteed learning rates.