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Traditional measures of smoothness often fail to provide accurate $L_p$-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified…
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…
Based on a plane-wave expansion of the observation data in quasi-planar multi-static scattering scenarios, an improved formalism for image creation utilizing back-projection in the spatial domain is derived. The underlying integral…
The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end a variant of Chernoff's product formula is…
The input to the distant representatives problem is a set of $n$ objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are…
In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable…
A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last…
This paper concerns a class of composite image reconstruction models for impluse noise removal, which is rather general and covers existing convex and nonconvex models proposed for reconstructing images with impluse noise. For this…
We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view of Birkhoff-James orthogonality and semi-inner-products. As an application of the present study, some distance formulae are…
The regression problem associated with finding a matrix approximation of the Koopman operator from data is considered. The regression problem is formulated as a convex optimization problem subject to linear matrix inequality (LMI)…
In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel…
Continuing earlier investigations, we analyze the convergence of operator splitting procedures combined with spatial discretization and rational approximations.
The forward-backward operator splitting algorithm is one of the most important methods for solving the optimization problem of the sum of two convex functions, where one is differentiable with a Lipschitz continuous gradient and the other…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function.…
The aim of this paper is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations of molecules. The Lieb functional was actually shown by…
Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown…
Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions $\varphi_j$ and a sequence of distributions/functions $\widetilde{\varphi}_j$. Error…
For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact…
In plug-and-play (PnP) regularization, the proximal operator in algorithms such as ISTA and ADMM is replaced by a powerful denoiser. This formal substitution works surprisingly well in practice. In fact, PnP has been shown to give…