scholarly journals The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jin Li ◽  
Xiuzhen Li
Keyword(s):  
Error Function ◽  
Quadrature Rule ◽  
Posteriori Error ◽  
Trapezoidal Rule ◽  
Boundary Element Methods ◽  
Posteriori Error Estimate ◽  
Order Of Accuracy ◽  
Hypersingular Integrals ◽  
Error Functional ◽  
The Error Functional

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

scholarly journals The order of convergence of an optimal quadrature formula with derivative in the space W(2,1)2

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3835-3844
Author(s):  
A.R. Hayotov ◽  
R.G. Rasulov
Keyword(s):  
Quadrature Formula ◽  
Order Of Convergence ◽  
Discrete Analog ◽  
Trapezoidal Rule ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
First Derivative ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The present work is devoted to extension of the trapezoidal rule in the space W(2,1)2. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of an integrand. Using the discrete analog of the operator d2/dx2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span{1,x,ex,e-x}. Finally, in the space W(2,1) 2 the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L(2)2 .

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Improving the Generalization Properties of Radial Basis Function Neural Networks

1991 ◽  
Vol 3 (4) ◽  
pp. 579-588 ◽  
Author(s):  
Chris Bishop
Keyword(s):  
Neural Networks ◽  
Radial Basis Function ◽  
Basis Function ◽  
Learning Algorithm ◽  
Error Functional ◽  
Radial Basis ◽  
The Error Functional ◽  
Network Mapping ◽  
Nonlinear Mappings

An important feature of radial basis function neural networks is the existence of a fast, linear learning algorithm in a network capable of representing complex nonlinear mappings. Satisfactory generalization in these networks requires that the network mapping be sufficiently smooth. We show that a modification to the error functional allows smoothing to be introduced explicitly without significantly affecting the speed of training. A simple example is used to demonstrate the resulting improvement in the generalization properties of the network.

scholarly journals divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations

2014 ◽  
Vol 7 (1) ◽  
pp. 225-241 ◽  
Author(s):  
A. Barth ◽  
J.-M. Beckers ◽  
C. Troupin ◽  
A. Alvera-Azcárate ◽  
L. Vandenbulcke
Keyword(s):  
Cost Function ◽  
Dimensional Analysis ◽  
Computational Cost ◽  
Posteriori Error ◽  
Cholesky Factorization ◽  
Ocean Current ◽  
Posteriori Error Estimate ◽  
Two Dimensional ◽  
A Posteriori ◽  
A Posteriori Error

Abstract. A tool for multidimensional variational analysis (divand) is presented. It allows the interpolation and analysis of observations on curvilinear orthogonal grids in an arbitrary high dimensional space by minimizing a cost function. This cost function penalizes the deviation from the observations, the deviation from a first guess and abruptly varying fields based on a given correlation length (potentially varying in space and time). Additional constraints can be added to this cost function such as an advection constraint which forces the analysed field to align with the ocean current. The method decouples naturally disconnected areas based on topography and topology. This is useful in oceanography where disconnected water masses often have different physical properties. Individual elements of the a priori and a posteriori error covariance matrix can also be computed, in particular expected error variances of the analysis. A multidimensional approach (as opposed to stacking two-dimensional analysis) has the benefit of providing a smooth analysis in all dimensions, although the computational cost is increased. Primal (problem solved in the grid space) and dual formulations (problem solved in the observational space) are implemented using either direct solvers (based on Cholesky factorization) or iterative solvers (conjugate gradient method). In most applications the primal formulation with the direct solver is the fastest, especially if an a posteriori error estimate is needed. However, for correlated observation errors the dual formulation with an iterative solver is more efficient. The method is tested by using pseudo-observations from a global model. The distribution of the observations is based on the position of the Argo floats. The benefit of the three-dimensional analysis (longitude, latitude and time) compared to two-dimensional analysis (longitude and latitude) and the role of the advection constraint are highlighted. The tool divand is free software, and is distributed under the terms of the General Public Licence (GPL) (http://modb.oce.ulg.ac.be/mediawiki/index.php/divand).

A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 1119-1139 ◽  
Author(s):  
RODOLFO ARAYA ◽  
ABNER H. POZA ◽  
ERNST P. STEPHAN
Keyword(s):  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Diffusion Reaction ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Advection Diffusion ◽  
Norm Of The Error ◽  
Reaction Problem

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

A Posteriori Error Analysis for the Weak Galerkin Method for Solving Elliptic Problems

2018 ◽  
Vol 15 (08) ◽  
pp. 1850075 ◽  
Author(s):  
Tie Zhang ◽  
Yanli Chen
Keyword(s):  
Elliptic Problems ◽  
Posteriori Error ◽  
Upper And Lower Bounds ◽  
Posteriori Error Estimate ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
A Posteriori Error ◽  
Galerkin Finite Element ◽  
Posteriori Error Analysis

In this paper, we study the a posteriori error estimate for weak Galerkin finite element method solving elliptic problems. A residual type error estimator is proposed and is proven to be reliable and efficient. This estimator provides global upper and lower bounds on the exact error in a discrete [Formula: see text]-norm. Numerical experiments are given to illustrate the effectiveness of the proposed error estimator.

Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate

2018 ◽  
Vol 26 (4) ◽  
pp. 493-499 ◽  
Author(s):  
Alexander S. Leonov ◽  
Alexander N. Sharov ◽  
Anatoly G. Yagola
Keyword(s):  
Error Estimate ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Posteriori Error ◽  
Theory Of Elasticity ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
Unknown Parameters ◽  
A Posteriori Error

Abstract This article presents the solution of a special inverse elastography problem: knowing vertical displacements of compressed biological tissue to find a piecewise constant distribution of Young’s modulus in an investigated specimen. Our goal is to detect homogeneous inclusions in the tissue, which can be interpreted as oncological. To this end, we consider the specimen as two-dimensional elastic solid, displacements of which satisfy the differential equations of the linear static theory of elasticity in the plain strain statement. The inclusions to be found are specified by parametric functions with unknown geometric parameters and unknown Young’s modulus. Reducing this inverse problem to the search for all unknown parameters, we solve it applying the modified method of extending compacts by V. K. Ivanov and I. N. Dombrovskaya. A posteriori error estimate is carried out for the obtained approximate solutions.

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