Global Error Estimates Research Articles

RKGHeM(4,4) technique with error control for connected component detector

The aim of this paper is focused on proposing a new versatile Runge-Kutta embedded fourth order four stage geometric heronian mean algorithm for simulating connected component detector (CCD) under cellular neural network (CNN) architecture. The mathematical expressions for local truncation error (LTE), global truncation error (GTE) and error estimates have been derived. Simulation results are presented for new RKGHeM(4,4) algorithm.

Mean‐Square Convergence of Drift‐Implicit One‐Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

A class of drift‐implicit one‐step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean‐square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean‐square convergence theory is illustrated by the stochastic θ‐methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean‐square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean‐square convergence of the stochastic θ‐methods and the balanced implicit methods are also new.

Decentralized Linear Motion Estimators for AUV Formations with Fixed Topologies

AbstractThis paper addresses the problem of decentralized state estimation in formations of Autonomous Underwater Vehicles (AUVs) with fixed topologies. In the envisioned scenario, each AUV in the formation estimates its own state relying only on locally available measurements and data communicated by neighboring agents, requiring lower computational and communication loads than centralized solutions. A method for designing local state observers featuring global error dynamics that converge asymptotically to zero is detailed, and an algorithm for improving its performance under stochastic disturbances and Gaussian uncertainties is presented. The proposed algorithm aims to minimize the H2 norm of the global estimation error dynamics, expressed as an optimization problem subject to Bilinear Matrix Inequality (BMI) constraints. To assess the performance of the solution, realistic simulation results are presented and discussed for several formation topologies.

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O ( h 6 ) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.

A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems

A code based on the two-stage Gauss formula (order four) for second-order initial value problems of a special type is developed. This code can be used to obtain a low- to medium-precision integration for a wide range of problems in the class of oscillatory type, Hamiltonian problems, and time-dependent partial differential equations discretized in space by finite differences or finite elements. The iteration process used in solving for the stage values of the Gauss formula, the selection of the initial step size, and the choice of an appropriate local error estimator for determining the step size change according to a particular tolerance specified by the user are studied. Moreover, a global error estimate and a dense output at equidistant points in the integration interval are supplied with the code. Numerical experiments and some comparisons with certain standard codes on relevant test problems are also given.

A posteriori error analysis of a domain decomposition algorithm for unilateral contact problem

In this paper, we consider a domain decomposition algorithm associated to a finite element method to approximate a unilateral frictionless contact problem between two elastic bodies. We present a global error estimator that takes into account of the error introduced by finite element analysis as well as the error introduced by the iterative resolution of the domain decomposition algorithm. The control of these error sources is a key point in order to introduce adaptive techniques and we propose error indicators that estimate the contribution of each source of error.

A simple yet effective a posteriori estimator for classical mixed approximation of Stokes equations

The implementation of quadratic velocity, linear pressure finite element approximation methods for the steady-state incompressible (Navier–)Stokes equations is addressed in this work. Three types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretisation error. Computational results suggest that the solution of local Poisson problems provides a cost-effective error estimation strategy, both from the perspective of accurate estimation of the global error and for the purpose of selecting elements for refinement within a contemporary self-adaptive refinement algorithm.

Distributed Estimation of Gauss - Markov Random Fields With One-Bit Quantized Data

We consider the problem of distributed estimation of a Gauss-Markov random field using a wireless sensor network (WSN), where due to the stringent power and communication constraints, each sensor has to quantize its data before transmission. In this case, the convergence of conventional iterative matrix-splitting algorithms is hindered by the quantization errors. To address this issue, we propose a one-bit adaptive quantization approach which leads to decaying quantization errors. Numerical results show that even with one bit quantization, the proposed approach achieves a superior mean square deviation performance (with respect to the global linear minimum mean-square error estimate) within a moderate number of iterations.

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction–diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε -weighted H 1 -norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε -weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

We present the asynchronous multi-domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary-value problem in hyperelasticity. Variational time integration schemes, based on the principle of minimal action within the Lagrangian framework, are constructed for the equation of motion and implemented into a variational finite element framework, which is systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly-incompressible materials. For efficient parallel computing, we use the dual domain decomposition method with local Lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains. The α-method for time discretization and the multi-domain spatial decomposition enable us to use different types of integrators (explicit vs. implicit) and different time steps on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. The energy conservation of our nonlinear, midpoint, asynchronous integration scheme is investigated using the Energy method, and both local and the global energy error estimates are derived. We illustrate the performance of proposed variational multi-domain time integrators by means of three examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we investigate energy conservation and stability. Finally, we apply the method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly with disparate material data of individual parts.

An adaptive version of Glimm's scheme

This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1(ℝ) for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered adhoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.

Sharply local pointwise a posteriori error estimates for parabolic problems

We prove pointwise a posteriori error estimates for semi- and fully-discrete finite element methods for approximating the solution u to a parabolic model problem. Our estimates may be used to bound the finite element error ∥u - u h ∥L ∞ (D), where D is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from D. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from D in order to ensure that local solution quality is not polluted by global effects.

An O(h 6 ) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h 2) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h 6) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.

Global error control in implicit parallel peer methods

Recently, Schmitt, Weiner, and Erdmann have proposed an efficient family of numerical methods termed Implicit Parallel Peer (IPP) methods. They are a subclass of s -stage general linear methods of order s – 1. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff differential equations. The special construction of IPP methods allows for a parallel implementation, which is advantageous in modern high-performance computation environment. In this paper we add one more useful functionality to IPP methods, i.e. automatic global error control. We show that the global error estimation developed by Kulikov and Shindin in multistep formulas is suitable for the methods of Schmitt, Weiner and Erdmann. Moreover, that global error estimation can be done in parallel. An algorithm of efficient stepsize selection is also discussed here.

Mean-square optimization of Monte Carlo algorithms for estimation of the solution to an integral equation of the second kind

The mean-square error minimization of the global histogram-type error estimate of the Monte Carlo method is performed. In the special case, the simulation of a Markov chain with transitional density proportional to the product of the absolute value of the original kernel and the averaging weight is optimal. In the general case, the transitional density minimizing the weighted sum of the variances of several functionals is basic. A sufficiently simple approximation to the asymptotic variant of such density is obtained. The minimax variant of the algorithm is also considered.

Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control

In this paper, we present variable-stepsize explicit parallel peer methods grounded in the interpolation idea. Approximation, stability, and convergence are studied in detail. In particular, we prove that some interpolating peer methods are stable on any variable mesh in practice. Double quasi consistency is utilized to introduce an efficient global error estimation formula in the numerical methods under discussion. The main advantage of these new adaptive schemes is the fact that the leading terms of their local and true errors coincide. Thus, controlling the local error of such methods by cheap standard techniques automatically regulates their global error as well. Numerical experiments of this paper support theoretical results presented below and illustrate how the new global error control concept works in practice. We also conduct a comparison with explicit ODE solvers in MATLAB.

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation

Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods. Here, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer methods constructed by Weiner, Schmitt, Podhaisky and Jebens. We present a sample of s -stage doubly quasi-consistent parallel explicit peer methods of order s − 1 when s = 3 . The notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this paper confirm clearly that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode.

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.

Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

AbstractThis study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super‐element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super‐element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error. Copyright © 2009 John Wiley & Sons, Ltd.