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.

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.

Tapio Helin, Stefan Kindermann, Jonatan Lehtonen et al.

Adaptive optics (AO) is a technology in modern ground-based optical telescopes to compensate the wavefront distortions caused by atmospheric turbulence. One method that allows to retrieve information about the atmosphere from telescope data is so-called SLODAR, where the atmospheric turbulence profile is estimated based on correlation data of Shack--Hartmann wavefront measurements. This approach relies on a layered Kolmogorov turbulence model. In this article, we propose a novel extension of the SLODAR concept by including a general non-Kolmogorov turbulence layer close to the ground with an unknown power spectral density. We prove that the joint estimation problem of the turbulence profile above ground simultaneously with the unknown power spectral density at the ground is ill-posed and propose three numerical reconstruction methods. We demonstrate by numerical simulations that our methods lead to substantial improvements in the turbulence profile reconstruction, compared to standard SLODAR-type approach. Also, our methods can accurately locate local perturbations in non-Kolmogorov power spectral densities.

Ronny Ramlau, Christoph Koutschan, Bernd Hofmann

In theory and practice of inverse problems, linear operator equations $Tx=y$ with compact linear forward operators $T$ having a non-closed range $\mathcal{R}(T)$ and mapping between infinite dimensional Hilbert spaces plays some prominent role. As a consequence of the ill-posedness of such problems, regularization approaches are required, and due to its unlimited qualification spectral cut-off is an appropriate method for the stable approximate solution of corresponding inverse problems. For this method, however, the singular system $\{σ_i(T),u_i(T),v_i(T)\}_{i=1}^\infty$ of the compact operator $T$ is needed, at least for $i=1,2,...,N$, up to some stopping index $N$. In this note we consider $n$-fold integration operators $T=J^n\;(n=1,2,...)$ in $L^2([0,1])$ occurring in numerous applications, where the solution of the associated operator equation is characterized by the $n$-th generalized derivative $x=y^{(n)}$ of the Sobolev space function $y \in H^n([0,1])$. Almost all textbooks on linear inverse problems present the whole singular system $\{σ_i(J^1),u_i(J^1),v_i(J^1)\}_{i=1}^\infty$ in an explicit manner. However, they do not discuss the singular systems for $J^n,\;n \ge 2$. We will emphasize that this seems to be a consequence of the fact that for higher $n$ the eigenvalues $σ^2_i(J^n)$ of the associated ODE boundary value problems obey transcendental equations, the complexity of which is growing with $n$. We present the transcendental equations for $n=2,3,...$ and discuss and illustrate the associated eigenfunctions and some of their properties.

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.

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.