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Regularization techniques for the numerical solution of inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of…
Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world…
Convex $\ell_1$ regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can…
The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in…
We introduce a general framework for the reconstruction of periodic multivariate functions from finitely many and possibly noisy linear measurements. The reconstruction task is formulated as a penalized convex optimization problem, taking…
Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the high dimensional setting. Only recently a few sparse recovery results have been established for some specific local…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…
Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel…
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…
Several generalizations of the traditional Tikhonov-Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers…
In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random…
This study presents the development of a spatially adaptive weighting strategy for Total Variation regularization, aimed at addressing under-determined linear inverse problems. The method leverages the rapid computation of an accurate…
We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the…
Explicit solutions to optimal control problems are rarely obtainable. Of particular interest are the explicit solutions derived for minimax problems, providing a framework to address adversarial conditions and uncertainty. This work…
The solution of inverse problems is of fundamental interest in medical and astronomical imaging, geophysics as well as engineering and life sciences. Recent advances were made by using methods from machine learning, in particular deep…
We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections…
We present an adaptive regularization scheme for optimizing composite energy functionals arising in image analysis problems. The scheme automatically trades off data fidelity and regularization depending on the current data fit during the…
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties…
The past several years have witnessed a surge of research investigating various aspects of sparse representations and compressed sensing. Most of this work has focused on the finite-dimensional setting in which the goal is to decompose a…
Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with…