Simon Hubmer

Simon Hubmer, Ekaterina Sherina, Andreas Neubauer et al.

We consider a problem of quantitative static elastography, the estimation of the Lamé parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lamé parameters from displacement data simulating a static elastography experiment are presented.

Simon Hubmer, Ronny Ramlau

In this paper, we consider Nesterov's Accelerated Gradient method for solving Nonlinear Inverse and Ill-Posed Problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide a convergence analysis for ill-posed problems of this method based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.

Simon Hubmer, Kim Knudsen, Changyou Li et al.

This paper considers the reconstruction problem in Acousto-Electrical Tomography, i.e., the problem of estimating a spatially varying conductivity in a bounded domain from measurements of the internal power densities resulting from different prescribed boundary conditions. Particular emphasis is placed on the limited angle scenario, in which the boundary conditions are supported only on a part of the boundary. The reconstruction problem is formulated as an optimization problem in a Hilbert space setting and solved using Landweber iteration. The resulting algorithm is implemented numerically in two spatial dimensions and tested on simulated data. The results quantify the intuition that features close to the measurement boundary are stably reconstructed and features further away are less well reconstructed. Finally, the ill-posedness of the limited angle problem is quantified numerically using the singular value decomposition of the corresponding linearized problem.

Lukas Weissinger, Simon Hubmer, Bernadett Stadler et al.

Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems. It is used to enhance the image quality of ground-based telescopes, such as for the Multiconjugate Adaptive Optics Relay For ELT Observations (MORFEO) instrument on the Extremely Large Telescope (ELT). To solve this problem, a singular-value decomposition (SVD) based approach has been proposed before. In this paper, we focus on the numerical implementation of the SVD-based Atmospheric Tomography with Fourier Domain Regularization Algorithm (SAFR) and its performance for Multi-Conjugate Adaptive Optics (MCAO) systems. The key features of the SAFR algorithm are the utilization of the FFT and the pre-computation of computationally demanding parts. Together, this yields a fast algorithm with less memory requirements than commonly used Matrix Vector Multiplication (MVM) approaches. We evaluate the performance of SAFR regarding reconstruction quality and computational expense in numerical experiments using the simulation environment COMPASS, in which we use an MCAO setup resembling the physical parameters of the MORFEO instrument of the ELT.

Helmut Gfrerer, Simon Hubmer, Stefan Kindermann et al.

In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via smooth approximations, which are inexact, or using non-smooth optimization techniques, which can often be numerically expensive, in particular for large-scale problems. Here, we present a numerically efficient minimization approach based on the recently proposed semismooth* Newton method, which employs a novel concept of graphical derivatives and exhibits locally superlinear convergence. The proposed approach is specifically tailored to TV regularization, suitable for large-scale inverse problems, and supported by strong mathematical convergence guarantees. Furthermore, we demonstrate its performance on two (large-scale) tomographic imaging problems and compare our results to those obtained via other state-of-the-art TV regularization approaches.

Simon Hubmer, Stefan Kindermann, Ekaterina Sherina

We consider the identification of a scalar coefficient in a PDE-based parameter estimation problem with contact constraints. The considered problem can be used as an idealized model of a membrane under forces, constrained by a barrier or indenter. More generally, it serves as a benchmark for the analysis of more complex contact problems and the development of corresponding reconstruction algorithms. In this paper, we discuss both the forward and inverse parameter estimation problems, as well as uniqueness and non-uniqueness issues caused by the contact constraints. Furthermore, we consider the design and implementation of reconstruction approaches which we test on numerical examples, illustrating both uniqueness and non-uniqueness as well as parameter identifyability.