Lam\'e Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

We consider a problem of quantitative static elastography, the estimation of the Lam\'e 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 Lam\'e parameters from displacement data simulating a static elastography experiment are presented.Comment: 29 page

Limited Angle Acousto-Electrical Tomography

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.Comment: 23 page

On Regularization via Frame Decompositions with Applications in Tomography

In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class of continuous regularization methods and derive convergence rates under both a-priori and a-posteriori parameter choice rules. Furthermore, we apply our derived results to a standard tomography problem based on the Radon transform.Comment: 30 pages, 6 figure

A projected Nesterov-Kaczmarz approach to stellar population-kinematic distribution reconstruction in Extragalactic Archaeology

In this paper, we consider the problem of reconstructing a galaxy's stellar population-kinematic distribution function from optical integral field unit measurements. These quantities are connected via a high-dimensional integral equation. To solve this problem, we propose a projected Nesterov-Kaczmarz reconstruction (PNKR) method, which efficiently leverages the problem structure and incorporates physical prior information such as smoothness and non-negativity constraints. To test the performance of our reconstruction approach, we apply it to a dataset simulated from a known ground truth density, and validate it by comparing our recoveries to those obtained by the widely used pPXF software.Comment: 34 pages, 8 figure

Localization of fixed dipoles at high precision by accounting for sample drift during illumination

Single molecule localization microscopy relies on the precise quantification of the position of single dye emitters in a sample. This precision is improved by the number of photons that can be detected from each molecule. It is therefore recommendable to increase illumination times for the recording process. Particularly recording at cryogenic temperatures dramatically reduces photobleaching and thereby allows a massive increase in illumination times to several seconds. As a downside, microscope instabilities may well introduce jitter during such long illuminations, deteriorating the localization precision. In this paper, we theoretically demonstrate that a parallel recording of fiducial marker beads together with a novel fitting approach accounting for the full drift trajectory allows for largely eliminating drift effects for drift magnitudes of several hundred nanometers per frame.Comment: 12 pages, 7 figure