Related papers: Sampling in BV-Type Spaces

In this paper, we address the random sampling problem for the class of Mellin band-limited functions BT which is concentrated on a bounded cube. It is established that any function in BT can be approximated by an element in a…

We present an elementary proof of the classical Beurling sampling theorem which gives a sufficient condition for sampling of multi-dimensional band-limited functions.

Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a…

We examine a fundamental problem that models various active sampling setups, such as network tomography. We analyze sampling of a multivariate normal distribution with an unknown expectation that needs to be estimated: in our setup it is…

We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as…

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…

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where…

Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which…

We study the inverse problem of deducing the dynamical characteristics (such as the potential field) of large systems from kinematic observations. We show that, for a class of steady-state systems, the solution is unique even with…

We develop a general optimization-theoretic framework for Bregman-Variational Learning Dynamics (BVLD), a new class of operator-based updates that unify Bayesian inference, mirror descent, and proximal learning under time-varying…

We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class…

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm…

Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…

We prove a sharp bound between sampling numbers and entropy numbers in the uniform norm for bounded convex sets of bounded functions.

We analyze a solution method for minimization problems over a space of $\mathbb{R}^d$-valued functions of bounded variation on an interval $I$. The presented method relies on piecewise constant iterates. In each iteration the algorithm…

On a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show that given a BV function, discarding a set of small $1$-capacity makes the function continuous outside its jump set and ``one-sidedly" continuous…

In metric measure spaces, we study boundary traces of BV functions in domains equipped with a doubling measure and supporting a Poincar\'e inequality, but possibly having a very large and irregular boundary. We show that the trace exists in…

A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned…