Related papers: A generalization of the Moreau-Yosida regularizati…
Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function $u(x)$ is \begin{eqnarray*}…
In a wide class of propagators regularized by the $\varepsilon$-metric [1], the $R$-operation is formulated. It is proved that the limit of renormalized Feynman integrals exists and is covariant. Possible applications in gravity are…
In the paper, we introduce the generalized convex function on fractal sets of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen inequality and…
We consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$, corrupted by…
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no…
We briefly review results on Colombeau type generalized solutions to the Cauchy problem for linear Schr\"odinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main…
Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method…
Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required.…
We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any…
In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces $A$ and $B$ of $C_0(X,E)$ and $C_0(Y,F)$ where $X$ and $Y$ are locally compact Hausdorff spaces and $E$ and $F$ are normed…
A detailed convex analysis-based formulation of density-functional theory for periodic systems in arbitrary dimensions is presented. The electron-electron interaction is taken to be of Yukawa type, harmonising with underlying function…
Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general…
We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( \alpha \right) }q=\left \vert K\right \vert…
We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where \(f\colon X\longrightarrow Y\) is twice continuously Fr\'echet-differentiable on Banach spaces and…
We study a standard method of regularization by projections of the linear inverse problem $Y=Af+\epsilon$, where $\epsilon$ is a white Gaussian noise, and $A$ is a known compact operator with singular values converging to zero with…
This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex…
For each integrability parameter $p \in (0,\infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$…
After reformulating Gromov's non-squeezing theorem as an area-inequality, we discuss a seemingly natural higher dimensional generalization.
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression…
In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In…