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The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions…
Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the…
We consider a circular deconvolution problem, in which the density $f$ of a circular random variable $X$ must be estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement error is…
This paper provides a framework to show the concentration of solutions $Y^*$ to convex minimizing problem where the objective function $\phi(X)(Y)$ depends on some random vector $X$ satisfying concentration of measure hypotheses. More…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a…
This paper introduces the Simultaneous assignment problem. Let us given a graph with a weight and a capacity function on its edges, and a set of its subgraphs along with a degree upper bound function for each of them. We are also given a…
A passive approximation problem is formulated where the target function is an arbitrary complex valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The…
The accuracy of density-functional theory (DFT) is determined by the quality of the approximate functionals, such as exchange-correlation in electronic DFT and the excess functional in the classical DFT formalism of fluids. The exact…
The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
In this paper we consider the problem of approximating function evaluations $f(\boldsymbol x_j)$ at given nonequispaced points $\boldsymbol x_j$, $j=1,\dots N$, of a bandlimited function from given values $\hat{f}(\boldsymbol k)$,…
Density-corrected density functional theory (DC-DFT) is enjoying substantial success in improving semilocal DFT calculations in a wide variety of chemical problems. This paper provides the formal theoretical framework and assumptions for…
In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such…
Large scale Density Functional Theory (DFT) based electronic structure calculations are highly time consuming and scale poorly with system size. While semi-empirical approximations to DFT result in a reduction in computational time versus…
The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it,…
Feedback particle filter (FPF) is a numerical algorithm to approximate the solution of the nonlinear filtering problem in continuous-time settings. In any numerical implementation of the FPF algorithm, the main challenge is to numerically…
Several questions of approximation theory are discussed: 1) can one approximate stably in $L^\infty$ norm $f^\prime$ given approximation $f_\delta, \parallel f_\delta - f \parallel_{L^\infty} < \delta$, of an unknown smooth function $f(x)$,…
This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance…
Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases…