scholarly journals Approximation properties of periodic multivariate quasi-interpolation operators

2021 ◽  
pp. 105631
Author(s):  
Yurii Kolomoitsev ◽  
Jürgen Prestin
Keyword(s):  
Approximation Properties ◽  
Interpolation Operators

scholarly journals Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast

2016 ◽  
Vol 16 (4) ◽  
pp. 579-603 ◽  
Author(s):  
Daniel Peterseim ◽  
Robert Scheichl
Keyword(s):  
Diffusion Coefficient ◽  
Orthogonal Decomposition ◽  
High Contrast ◽  
Approximation Properties ◽  
New Approach ◽  
Multiscale Problems ◽  
Optimal Convergence ◽  
Discretization Schemes ◽  
Interpolation Operators ◽  
New Framework

AbstractWe present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of ${H^{1}}$ into the image and the kernel of some novel stable quasi-interpolation operators with local $L^{2}$-approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretization schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.

scholarly journals Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

2020 ◽  
Vol 30 (09) ◽  
pp. 1809-1855
Author(s):  
Daniele A. Di Pietro ◽  
Jérôme Droniou ◽  
Francesca Rapetti
Keyword(s):  
Finite Element ◽  
Commutative Diagram ◽  
Three Dimensional ◽  
Computer Implementation ◽  
Approximation Properties ◽  
Arbitrary Degree ◽  
Fully Discrete ◽  
De Rham ◽  
Interpolation Operators ◽  
Suitable Approximation

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element (FE) spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete [Formula: see text]-products completes the exposition.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Approximation properties of Szász type operators involving Charlier polynomials

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 479-487
Author(s):  
Didem Arı
Keyword(s):  
Asymptotic Formula ◽  
Weighted Space ◽  
Approximation Properties ◽  
Charlier Polynomials

In this paper, we give some approximation properties of Sz?sz type operators involving Charlier polynomials in the polynomial weighted space and we give the quantitative Voronovskaya-type asymptotic formula.

scholarly journals Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Valdete Loku ◽  
Naim L. Braha ◽  
Toufik Mansour ◽  
M. Mursaleen
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Approximation Properties ◽  
Type Result ◽  
Szász Operators ◽  
Sheffer Polynomials

AbstractThe main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.

Sign in / Sign up

Export Citation Format

Share Document