Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a ...regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.
In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank ...approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nyström method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.
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Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits ...of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.
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We consider the problem of blob detection for uncertain images, such as images that have to be inferred from noisy measurements. Extending recent work motivated by astronomical applications, we ...propose an approach that represents the uncertainty in the position and size of a blob by a region in a three-dimensional scale space. Motivated by classic tube methods such as the taut-string algorithm, these regions are obtained from level sets of the minimizer of a total variation functional within a high-dimensional tube. The resulting non-smooth optimization problem is challenging to solve, and we compare various numerical approaches for its solution and relate them to the literature on constrained total variation denoising. Finally, the proposed methodology is illustrated on numerical experiments for deconvolution and models related to astrophysics, where it is demonstrated that it allows to represent the uncertainty in the detected blobs in a precise and physically interpretable way.
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In this paper we study the attenuated X-ray transform of 2-tensors supported in convex bounded subsets with sufficiently smooth boundary in the Euclidean plane. We characterize its range and ...reconstruct all possible 2-tensors yielding identical X-ray data. The characterization is in terms of a Hilbert-transform associated with A-analytic maps in the sense of Bukhgeim.
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We provide computationally generated dataset simulating propagation of ultrasonic waves in viscous tissues in two and three dimensional domains. The dataset contains physical parameters of a human ...breast with a high-contrast inclusion, the acquisition setup with positions of sources and receivers, and the associated pressure-wave data at ultrasonic frequencies. We simulated the wave propagation based on seven different viscous models using the physical parameters of the breast. Furthermore, different choices of conditions for the medium's boundaries are given, namely absorbing and reflecting boundaries. The dataset allows to evaluate the performance of reconstruction methods for ultrasound imaging under attenuation model uncertainty, that is, when the precise attenuation law that characterizes the medium is unknown. In addition, the dataset enables to evaluate the robustness of inverse scheme in the context of reflecting boundary conditions where multiple reflections illuminate the sample, and/or the performance of data-processing to suppress these multiple reflections.
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We present a family of
variational regularization methods for solving
problems,
where the solutions are functions with range in a closed subset of the Euclidean space, for example if the
solution ...only attains values in an embedded sub-manifold. Recently, in R. Ciak, M. Melching and O. Scherzer,
Regularization with metric double integrals of functions with values in a set of vectors,
J. Math. Imaging Vision 61 2019, 6, 824–848, such regularization methods
have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and
inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.
The goal of this book is three-fold: it describes the basics of model order reduction and related aspects. In numerical linear algebra, it covers both general and more specialized model order ...reduction techniques for linear and nonlinear systems, and it discusses the use of model order reduction techniques in a variety of practical applications. The book contains many recent advances in model order reduction, and presents several open problems for which techniques are still in development. It will serve as a source of inspiration for its readers, who will discover that model order reduction is a very exciting and lively field.
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Single molecule localization microscopy (SMLM) has enormous potential for resolving subcellular structures below the diffraction limit of light microscopy: Localization precision in the low digit ...nanometer regime has been shown to be achievable. In order to record localization microscopy data, however, sample fixation is inevitable to prevent molecular motion during the rather long recording times of minutes up to hours. Eventually, it turns out that preservation of the sample's ultrastructure during fixation becomes the limiting factor. We propose here a workflow for data analysis, which is based on SMLM performed at cryogenic temperatures. Since molecular dipoles of the fluorophores are fixed at low temperatures, such an approach offers the possibility to use the orientation of the dipole as an additional information for image analysis. In particular, assignment of localizations to individual dye molecules becomes possible with high reliability. We quantitatively characterized the new approach based on the analysis of simulated oligomeric structures. Side lengths can be determined with a relative error of less than 1% for tetramers with a nominal side length of 5 nm, even if the assumed localization precision for single molecules is more than 2 nm.
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