Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. 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 problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

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.     

Parallel start for explicit parallel two-step peer methods

Explicit parallel two-step peer methods use s stages with essentially identical properties. They are quite efficient in solving standard nonstiff initial value problems and may obtain a parallel speed-up near s on s processors for expensive problems. The two-step structure requires s???1 initial approximations which have been computed by one-step methods in earlier versions. We now present a self-contained starting procedure using parallel Euler steps in the initial interval. Low order error terms introduced by this step are eliminated by special coefficient sets increasing the order to s after s???2 time steps. An estimate for the initial stepsize is discussed, as well. Parallel OpenMP experiments with realistic problems demonstrate the efficiency compared to standard codes.     

W-methods in optimal control

This paper addresses consistency and stability of W-methods up to order three for nonlinear ODE-constrained control problems with possible restrictions on the control. The analysis is based on the transformed adjoint system and the control uniqueness property. These methods can also be applied to large-scale PDE-constrained optimization, since they offer an efficient way to compute gradients of the discrete objective function.     

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

In this paper it is shown that the local discretization error ofs-stage singly-implicit methods of orderp can be estimated by embedding these methods intos-stage two-step Runge-Kutta methods of orderp+1, wherep=s orp=s+1. These error estimates do not require any extra evaluations of the right hand side of the differential equations. This is in contrast with the error estimation schemes based on embedded pairs of two singly-implicit methods proposed by Burrage.The work of A. Bellen and M. Zennaro was supported by the CNR and MPI. The work of Z. Jackiewicz was supported by the CNR and by the NSF under grant DMS-8520900.     

Global error estimation with adaptive explicit Runge-Kutta methods

Users of locally-adaptive software for initial value ordinarydifferential equations are likely to be concerned with globalerrors. At the cost of extra computation, global error estimationis possible. Zadunaisky's method and ‘solving for theerror estimate’ are two techniques that have been successfullyincorporated into Runge-Kutta algorithms. The standard erroranalysis for these techniques, however, does not take accountof the stepsize selection mechanism. In this paper, some newresults are presented which, under suitable assumptions showthat these techniques are asymptotically valid when used withan adaptive, variable stepsize algorithm—the global errorestimate reproduces the leading term of the global error inthe limit as the error tolerance tends to zero. The analysisis also applied to Richardson extrapolation (step halving).Numerical results are provided for the technique of solvingfor the error estimate with several Runge-Kutta methods of Dormand,Lockyer, McGorrigan and Prince.     

Global error estimation for the implicit trapezoidal rule

We find an accurate recursion relation for the global error in the trapezoidal rule. The case of the simple pendulum is used to illustrate the loss of numerical accuracy that results from the use of a phase lag in the formula.     

Global error estimation for index-1 and -2 DAEs

In this paper, we consider the extension of three classical ODE estimation techniques (Richardson extrapolation, Zadunaisky's technique and solving for the correction) to DAEs. Their convergence analysis is carried out for semi-explicit index-1 DAEs solved by a wide set of Runge-Kutta methods. Experimentation of the estimation techniques with RADAU5 is also presented: their behaviour for index-1 and -2 problems, and for variable step size integration is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global property of error density estimation in nonlinear autoregressive time series models

This paper considers estimation of the error density function in nonlinear autoregressive stationary time series regression model. The asymptotic distribution of the maximum of a suitably normalized deviation of the density estimator from the expectation of the kernel error density (based on the true error) is obtained to be the same as in the case of the one sample set up, which is given in Bickel and Rosenblatt (Ann Stat 6:1071–1095, 1973).     

Reliable a posteriori error estimation for state-constrained optimal control

We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter.     

线性常微分方程的全局误差估计和优化求解方法

线性常微分方程初值问题求解在许多应用中起着重要作用.目前,已存在很多的数值方法和求解器用于计算离散网格点上的近似解,但很少有对全局误差(global error)进行估计和优化的方法.本文首先通过将离散数值解插值成为可微函数用来定义方程的残差;再给出残差与近似解的关系定理并推导出全局误差的上界;然后以最小化残差的二范数为目标将方程求解问题转化为优化求解问题;最后通过分析导出矩阵的结构,提出利用共轭梯度法对其进行求解.之后将该方法应用于滤波电路和汽车悬架系统等实际问题.实验分析表明,本文估计方法对线性常微分方程的初值问题的全局误差具有比较好的估计效果,优化求解方法能够在不增加网格点的情形下求解出线性常微分方程在插值解空间中的全局最优解.     

Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods

Automatic global error control of numerical schemes is examined. A new approach to this problem is presented. Namely, the problem is reformulated so that the global error is controlled by the numerical method itself rather than by the user. This makes it possible to find numerical solutions satisfying various accuracy requirements in a single run, which so far was considered unrealistic. On the other hand, the asymptotic equality of local and global errors, which is the basic condition of the new method for efficiently controlling the global error, leads to the concept of double quasi-consistency. This requirement cannot be satisfied within the classical families of numerical methods. However, the recently proposed peer methods include schemes with this property. There exist computational procedures based on these methods and polynomial interpolation of fairly high degree that find the numerical solution in a single run. If the integration stepsize is sufficiently small, the error of this solution does not exceed the prescribed tolerance. The theoretical conclusions of this paper are supported by the numerical results obtained for test problems with known solutions.     

A parallel approach to improvement and estimation of the approximate optimal control

In this paper the method for computing a priori estimates of the approximate optimal control is considered. These estimates provide us with information about the quality of the approximate optimal solution obtained by applying the improvement control procedure. The method is implemented in the form of a parallel algorithm. This algorithm is an essential part of the developed software package intended for optimization of controllable dynamical systems. We also consider the scalability of the parallel algorithm in the OpenTS parallel programming system for chemical and biochemical engineering problems.     

Accurate error estimation in CG

Numerical Algorithms - In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations Ax = b...     

Functional approach to a posteriori error estimation for elliptic optimal control problems with distributed control

A new approach to the a posteriori analysis of distributed optimal control problems is presented. The approach is based on functional type a posteriori estimates that provide computable and guaranteed bounds of errors for any conforming approximations of a boundary value problem. Computable two-sided a posteriori estimates for the cost functional and estimates for approximations of the state and control functions are derived. Numerical results illustrate the efficiency of the approach. Bibliography: 35 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 3–14     

A posteriori error estimation and error control for finite element approximations of the time-dependent Navier–Stokes equations

We present an approach to estimate numerical errors in finite element approximations of the time-dependent Navier–Stokes equations along with a strategy to control these errors. The error estimators and the error control procedure are based on the residuals of the Navier–Stokes equations, which are shown to be comparable to error components in the velocity variable. The present methodology applies to the estimation of numerical errors due to the spatial discretization only. Its performance is demonstrated for two-dimensional channel flows past a cylinder in the periodic regime.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

The convergence ball and error analysis of the two-step Secant method

Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order, we obtain an estimate of the radius of the convergence ball for the two-step secant method. Moreover, we also provide an error estimate that matches the convergence order of the two-step secant method. At last, we give an application of the proposed theorem.     

Stability and error of the variable two-step BDF for semilinear parabolic problems

The temporal discretisation of a moderate semilinear parabolic problem in an abstract setting by the two-step backward differentiation formula with variable step sizes is analysed. Stability as well as optimal smooth data error estimates are derived if the ratios of adjacent step sizes are bounded from above by 1.91.