scholarly journals Convergence rates of derivatives of a family of barycentric rational interpolants

2011 ◽  
Vol 61 (9) ◽  
pp. 989-1000 ◽  
Author(s):  
Jean-Paul Berrut ◽  
Michael S. Floater ◽  
Georges Klein
Keyword(s):  
Convergence Rates ◽  
Rational Interpolants ◽  
Derivatives Of

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

scholarly journals New error bounds for Laplace approximation via Stein's method

2021 ◽  
Author(s):  
Robert Gaunt
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Laplace Approximation ◽  
Stein’S Method ◽  
Independent Random Variables ◽  
Stein's Method ◽  
Smooth Test ◽  
Stein Equation ◽  
Technical Advances ◽  
Derivatives Of

We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren \cite {pike} for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

Moving Finite Element Methods for a System of Semi-Linear Fractional Diffusion Equations

2016 ◽  
Vol 8 (6) ◽  
pp. 911-931 ◽  
Author(s):  
Jingtang Ma ◽  
Zhiqiang Zhou
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Convergence Rates ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Predator Prey ◽  
Fractional Diffusion Equations ◽  
Predator Prey Models ◽  
Derivatives Of ◽  
Moving Finite Element

AbstractThis paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-prey models by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than two. Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved. Numerical examples are carried out to confirm the theoretical findings. Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.

scholarly journals The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
J. F. Wang ◽  
S. Y. Hao ◽  
Y. M. Cheng
Keyword(s):  
Boundary Value Problems ◽  
Convergence Rates ◽  
Boundary Value ◽  
Weak Form ◽  
Point Boundary ◽  
Element Free Galerkin Method ◽  
Original Solution ◽  
Element Free Galerkin ◽  
Element Free ◽  
Derivatives Of

The interpolating moving least-squares (IMLS) method is discussed in detail, and a simpler formula of the shape function of the IMLS method is obtained. Then, based on the IMLS method and the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method for two-point boundary value problems is presented. The IEFG method has high computing speed and precision. Then error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories show that, if the original solution is sufficiently smooth and the order of the basis functions is big enough, the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, two selected numerical examples are given to confirm the theories.

Convergence rates of derivatives of Floater–Hormann interpolants for well-spaced nodes

2017 ◽  
Vol 116 ◽  
pp. 108-118 ◽  
Author(s):  
Emiliano Cirillo ◽  
Kai Hormann ◽  
Jean Sidon
Keyword(s):  
Convergence Rates ◽  
Derivatives Of

scholarly journals CONVERGENCE RATES FOR ILL-POSED INVERSE PROBLEMS WITH AN UNKNOWN OPERATOR

2010 ◽  
Vol 27 (3) ◽  
pp. 522-545 ◽  
Author(s):  
Jan Johannes ◽  
Sébastien Van Bellegem ◽  
Anne Vanhems
Keyword(s):  
Convergence Rates ◽  
Regression Function ◽  
Rates Of Convergence ◽  
Upper Bounds ◽  
Structural Function ◽  
Unified Framework ◽  
Nonparametric Function ◽  
Ill Posed ◽  
Linear Transform ◽  
Derivatives Of

This paper studies the estimation of a nonparametric functionϕfrom the inverse problemr=Tϕgiven estimates of the functionrand of the linear transformT. We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The two assumptions quantify the prior regularity ofϕand the prior link existing betweenTand the Hilbert scale. The approach provides a unified framework that allows us to compare various sets of structural assumptions found in the econometric literature. Moreover, general upper bounds are also derived for the risk of the estimator of the structural functionϕas well as that of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Two important applications are also studied. The first is the blind nonparametric deconvolution on the real line, and the second is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.

scholarly journals Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

2019 ◽  
Vol 53 (3) ◽  
pp. 1061-1082
Author(s):  
Beichuan Deng ◽  
Jiwei Zhang ◽  
Zhimin Zhang
Keyword(s):  
Collocation Method ◽  
Fractional Derivatives ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Interpolation Polynomial ◽  
Eigenvalue Method ◽  
Convergence And Superconvergence ◽  
The One ◽  
Derivatives Of ◽  
Hermite Interpolations

In this paper, we study the theory of convergence and superconvergence for integer and fractional derivatives of the one-point and two-point Hermite interpolations. When considering the integer-order derivatives, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are O(N−2) and O(N−1.5) better than the global rates for the one-point and two-point interpolations, respectively. Here N represents the degree of the interpolation polynomial. It is proved that the αth fractional derivative of (u−uN), with k<α<k+1, is bounded by its (k+1) th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann–Liouville derivatives. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.

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