Ronny Ramlau

Ronny Ramlau, Lothar Reichel

Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.

Daniel Gerth, Andreas Hofinger, Ronny Ramlau

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is a crucial part. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.

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.

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.

Simon Hubmer, Lukas Weissinger, Ronny Ramlau et al.

In this paper, we consider Fourier phase retrieval from differential intensity measurements, i.e., the problem of determining the phase of a complex-valued function from a series of intensity measurements differing only by slight modulations in Fourier domain. These modulations may be induced by optical elements such as prisms or phase plates, or via spatial-light modulators. Generalizing the principles behind the transport of intensity equation, we show that given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics. Numerical experiments illustrate the usefulness of our proposed approach.