scholarly journals Stabilised explicit Adams-type methods

2021 ◽  
pp. 82-98
Author(s):  
Vasily I. Repnikov ◽  
Boris V. Faleichik ◽  
Andrew V. Moisa
Keyword(s):  
Numerical Experiments ◽  
Step Method ◽  
Type Method ◽  
Constrained Optimisation ◽  
First Order ◽  
Error Constant ◽  
Stability Intervals ◽  
Multi Step Methods ◽  
First Order Methods ◽  
Accuracy And Stability

In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.

Assessing Medical Evidence

2018 ◽  
Author(s):  
Jacob Stegenga
Keyword(s):  
Quality Assessment ◽  
Randomized Trials ◽  
Empirical Studies ◽  
Allocation Concealment ◽  
Assessment Tools ◽  
Medical Evidence ◽  
Empirical Methods ◽  
First Order ◽  
First Order Methods

Medical scientists employ ‘quality assessment tools’ to assess evidence from medical research, especially from randomized trials. These tools are designed to take into account methodological details of studies, including randomization, subject allocation concealment, and other features of studies deemed relevant to minimizing bias. There are dozens of such tools available. They differ widely from each other, and empirical studies show that they have low inter-rater reliability and low inter-tool reliability. This is an instance of a more general problem called here the underdetermination of evidential significance. Disagreements about the quality of evidence can be due to different—but in principle equally good—weightings of the methodological features that constitute quality assessment tools. Thus, the malleability of empirical research in medicine is deep: in addition to the malleability of first-order empirical methods, such as randomized trials, there is malleability in the tools used to evaluate first-order methods.

scholarly journals A General Framework for Decentralized Optimization With First-Order Methods

2020 ◽  
Vol 108 (11) ◽  
pp. 1869-1889
Author(s):  
Ran Xin ◽  
Shi Pu ◽  
Angelia Nedic ◽  
Usman A. Khan
Keyword(s):  
General Framework ◽  
Decentralized Optimization ◽  
First Order ◽  
First Order Methods

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

First-order methods of smooth convex optimization with inexact oracle

2013 ◽  
Vol 146 (1-2) ◽  
pp. 37-75 ◽  
Author(s):  
Olivier Devolder ◽  
François Glineur ◽  
Yurii Nesterov
Keyword(s):  
Convex Optimization ◽  
Smooth Convex ◽  
First Order ◽  
Inexact Oracle ◽  
Smooth Convex Optimization ◽  
First Order Methods

scholarly journals FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

2020 ◽  
Vol 4 (3) ◽  
pp. 313-322
Author(s):  
Sunday Obomeviekome Imoni ◽  
D. I. Lanlege ◽  
E. M. Atteh ◽  
J. O. Ogbondeminu
Keyword(s):  
Basis Function ◽  
Initial Value Problems ◽  
Hermite Polynomials ◽  
Multistep Method ◽  
Linear Multistep Method ◽  
Initial Value ◽  
Numerical Examples ◽  
First Order ◽  
Block Scheme ◽  
Multi Step Methods

ABSTRACT In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals First-Order Methods for Convex Optimization

2021 ◽  
pp. 100015
Author(s):  
Pavel Dvurechensky ◽  
Shimrit Shtern ◽  
Mathias Staudigl
Keyword(s):  
Convex Optimization ◽  
First Order ◽  
First Order Methods
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