Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

An approximation theory by bandlimited functions (≡ Paley-Wiener functions) on Riemannian manifolds of bounded geometry is developed. Based on this theory multiscale approximations to smooth functions in Sobolev and Besov spaces on manifolds are obtained. The results have immediate applications to the filtering, denoising and approximation and compression of functions on manifolds. There exists applications to problems arising in data dimension reduction, image processing, computer graphics, visualization and learning theory.      

Sufficient conditions for ε-minimum

The paper contains sufficient conditions for a point x ∈ U, function f X→ R ∪ ∪ {-∞,∞}, and ε≧ 0 to hold the inequality f(x) ≦ inf f=ε on the subset U of a topological vector space X.     

N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

Continuous ε-selection theorems for almost lower semicontinuous multifunctions and applications

Continuous ε-selection theorems for almost lower semicontinuous multifunctions with decomposable values and some of their applications are presented. In particular, the existence of approximation solutions of differential inclusions with almost lower semicontinuous right-hand sides is obtained.     

The ε-inclusion condition for N-compacta

Criteria for NED-sets and N-compacta generating normal domains are given in terms of -inclusion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 154–161, 1991.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ε-approximations

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (?⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω??³. Bibliography: 25 titles.     

Subdifferentials of Distance Functions, Approximations and Enlargements

In this work, we study some subdifferentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set.     

On ε-spectra and stability radii

Techniques of Krylov subspace iterations play an important role in computing ε-spectra of large matrices. To obtain results about the reliability of this kind of approximations, we propose to compare the position of the ε-spectrum of A with those of its diagonal submatrices. We give theoretical results which are valid for any block decomposition in four blocks, A₁₁,A₁₂,A₂₁,A₂₂. We then illustrate our results by numerical experiments. The same kind of problem arises when we compute the stability radius of a large matrix. In that context, we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices.     

Approximations of e and π: an exploration

ABSTRACT

Fractional approximations of e and π are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.     

Error Bounds for Glimm Difference Approximations for Scalar Conservation Laws Without Convexity

In last century, D. Hoff and J. Smoller derived the error bounds for the Glimm difference approximations of the solutions to scalar conservation laws with convexity. Our work is to extend the corresponding result of them to the case without convexity.     

Error Estimates for Rayleigh–Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation

Error estimates are derived for the computation of eigenvalues and eigenvectors of infinite tridiagonal matrices by the Rayleigh–Ritz method. The results are applied to the Mathieu and spheroidal wave equation.     

Particular rules for the vector ε-algorithm

Summary The aim of this paper is to give a singular rule for the vector -algorithm. This singular rule avoids a too rapid growth of rounding errors when applying the vector -algorithm. Another rule is given which works when two adjacent vectors are equal.     

Interior Estimates for the First-Order Differences for Finite-Difference Approximations for Elliptic Bellman’s Equations

We establish interior estimates for the first-order finite differences of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations.     

Pseudo-spectral Approximations for a Class of the Kdv-Burgers Type Equation

In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev‘s inequalities and the bounded extensive method of the nonlinear function. Finally, the numerical examples are proposed.     

Padé–Jacobi Filtering for Spectral Approximations of Discontinuous Solutions

Spectral type methods for the discretization of partial differential equations rely on the approximation of the solution by polynomials of high degree. These methods are proven, both theoretically and numerically, to be of infinite order of accuracy. This infinite order is achieved if the solution is very regular. On the other hand, the Gibbs phenomenon prevents – a priori – the good convergence if the solution is discontinuous. Nevertheless, for systems of conservation laws, the spectral vanishing viscosity method leads to numerical solutions that are spectrally close to the projection of the exact solution on the set of polynomials. The idea is then to postprocess the numerical solution in order to extract pertinent physical information. The aim of this paper is to propose and analyse such a postprocessing method based on rational approximants that allows to circumvent the Gibbs phenomenon and can be used as an acceleration device for spectral numerical solution.     

Approximations for Solutions of Lévy-Type Stochastic Differential Equations

Abstract

The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for Lévy-type stochastic differential equation. In particular, the article generalizes the results from [2 Gardoń , A. 2004 . The order of approximations for solutions of Ito-type stochastic differential equations with jumps . Stoch. Anal. Appl. 22 ( 3 ): 679 – 699 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 5 Kloeden , P.E. , and Platen , E. 1995 . Numerical Solutions of Stochastic Differential Equations . Springer-Verlag , Berlin . [Google Scholar]]. The Euler and the Milstein schemes are shown for finite and infinite Lévy measure.