Weight optimal order of convergence cubature formulas in Sobolev space

2021 ◽  
Author(s):  
Ozodjon Jalolov
Keyword(s):  
Sobolev Space ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Cubature Formulas

scholarly journals Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Malik Zaka Ullah ◽  
A. S. Al-Fhaid ◽  
Fayyaz Ahmad
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Optimal Order ◽  
Optimal Order Of Convergence

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=25-1) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

scholarly journals On the Acceleration of the Multi-Level Monte Carlo Method

2015 ◽  
Vol 52 (2) ◽  
pp. 307-322 ◽  
Author(s):  
Kristian Debrabant ◽  
Andreas Röβler
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Weak Approximation ◽  
Mean Square ◽  
Optimal Order Of Convergence ◽  
Computational Costs ◽  
The Mean ◽  
Multi Level

The multi-level Monte Carlo method proposed by Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor (p / α)2 if weak approximation methods of orders α and p are applied in the case of computational costs growing with the same order as the variances decay.

scholarly journals Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 164
Author(s):  
Moin-ud-Din Junjua ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Nonlinear Equation ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Science And Engineering ◽  
Optimal Order Of Convergence ◽  
Van Der Waals Equation ◽  
First Order ◽  
Root Finding ◽  
Derivative Free ◽  
Inverse Interpolation

Finding a simple root for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest due to its wide applications in many fields of science and engineering. Newton’s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.

Convergence Analysis of Discontinuous Finite Volume Methods for Elliptic Optimal Control Problems

2016 ◽  
Vol 13 (02) ◽  
pp. 1640012 ◽  
Author(s):  
Ruchi Sandilya ◽  
Sarvesh Kumar
Keyword(s):  
Order Of Convergence ◽  
Optimal Control Problems ◽  
Finite Volume Methods ◽  
Optimal Order ◽  
Control Problems ◽  
State Variables ◽  
Optimal Order Of Convergence ◽  
Piecewise Constant ◽  
Adjoint State ◽  
Variational Discretization

In this paper, we discuss the convergence analysis of discontinuous finite volume methods applied to distribute the optimal control problems governed by a class of second-order linear elliptic equations. In order to approximate the control, two different methodologies are adopted: one is the method of variational discretization and second is piecewise constant discretization technique. For variational discretization method, optimal order of convergence in the [Formula: see text]-norm for state, adjoint state and control variables is derived. Moreover, optimal order of convergence in discrete [Formula: see text]-norm is also derived for state and adjoint state variables. Whereas, for piecewise constant approximation of control, first order convergence is derived for control, state and adjoint state variables in the [Formula: see text]-norm. In addition to that, optimal order of convergence in discrete [Formula: see text]-norm is derived for state and adjoint state variables. Also, some numerical experiments are conducted to support the derived theoretical convergence rate.

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850010 ◽  
Author(s):  
Janak Raj Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
Derivative Free ◽  
Local Convergence Analysis ◽  
Theoretical Results

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

Nonpolynomial collocation approximation of solutions to fractional differential equations

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Neville Ford ◽  
M. Morgado ◽  
Magda Rebelo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Error Analysis ◽  
Convergence Analysis ◽  
Fractional Differential Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Fractional Differential ◽  
Polynomial Collocation

AbstractWe propose a non-polynomial collocation method for solving fractional differential equations. The construction of such a scheme is based on the classical equivalence between certain fractional differential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional differential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.

scholarly journals The saturation phenomena for Tikhonov regularization

1983 ◽  
Vol 35 (2) ◽  
pp. 254-262 ◽  
Author(s):  
C. W. Groetsch ◽  
J. T. King
Keyword(s):  
Tikhonov Regularization ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Operator Equations ◽  
Optimal Order Of Convergence

AbstractThis paper is concerned with a characterization of the optimal order of convergence of Tikhonov regularization for first kind operator equations in terms of the “smoothness” of the data.

Sign in / Sign up

Export Citation Format

Share Document