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Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex…
Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework…
The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
We analytically solve for the time dependent solutions of various density evolution models. With specific forms of the diffusion, drift and sink coefficients, the eigenfunctions can be expressed in terms of hypergeometric functions. We…
The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. It consists of computing…
We study numerical integration of functions $f: \mathbb{R}^{s} \to \mathbb{R}$ with respect to a probability measure. By applying the corresponding inverse cumulative distribution function, the problem is transformed into integrating an…
We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational.…
We compute the rate of convergence of forward, backward and central finite difference $\theta$-schemes for linear PDEs with an arbitrary odd order spatial derivative term. We prove convergence of the first or second order for smooth and…
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of…
The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work…
In this paper, we consider derivative free optimization problems, where the objective function is smooth but is computed with some amount of noise, the function evaluations are expensive and no derivative information is available. We are…
A set of recursive relations satisfied by Selberg-type integrals involving monomial symmetric polynomials are derived, generalizing previously known results. These formulas provide a well-defined algorithm for computing Selberg-Schur…
We consider the problem of estimating the mixing density $f$ from $n$ i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast with finite mixtures models, here the distribution of the…
Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…
This paper addresses the study of derivative-free smooth optimization problems, where the gradient information on the objective function is unavailable. Two novel general derivative-free methods are proposed and developed for minimizing…
We look for spectral type differential equations for the generalized Jacobi polynomials and for the Sobolev-Laguerre polynomials. We use a method involving computeralgebra packages like Maple and Mathematica and we will give some…
This paper is devoted to the study of generalized differentiation properties of the infimal convolution. This class of functions covers a large spectrum of nonsmooth functions well known in the literature. The subdifferential formulas…