On global error estimation and control for initial value problems

This paper addresses global error estimation and control for initial value problems for ordinary differential equations. The focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation. Control is achieved through tolerance proportionality. Both approaches are found to work well and to enable estimation and control in a reliable manner. However, the novel approach is not found to be competitive with the classical approach, mainly because of its huge storage demand for large problems

Efficient goal-oriented global error estimation for BDF-type methods using discrete adjoints

This thesis develops estimation techniques for the global error that occurs during the approximation of solutions of Initial Value Problems (IVPs) on given intervals by multistep integration methods based on Backward Differentiation Formulas (BDF). To this end, discrete adjoints obtained by adjoint Internal Numerical Differentiation (IND) of the nominal integration scheme are used. For this purpose, a bridge between BDF methods and Petrov-Galerkin Finite Element (FE) methods is built by a novel functional-analytic framework. Goal-oriented global error estimators are derived in analogy to the Dual Weighted Residual methodology in Galerkin methods for Partial Differential Equations. Their asymptotic behavior, their accuracy in BDF methods with variable order and stepsize as well as their applicability for global error control are investigated. The novel results presented in this thesis include: i) a functional-analytic framework for IVPs in Ordinary Differential Equations (ODEs) in the Banach space of continuously differentiable functions. This framework is needed since the classical Hilbert space setting is not suitable to analyze the relation between the discrete values of the adjoint IND scheme and the solution of the adjoint IVP. The new framework gives rise to the definition of weak solutions of adjoint IVPs. ii) a Petrov-Galerkin FE discretization of the function spaces that allows to transform the variational formulations of the IVP and of its adjoint IVP into finite dimensional problems. The equivalence of these finite dimensional problems to BDF methods with variable but prescribed order and stepsize and their adjoint IND schemes is shown. Thus, the FE approximation of the weak adjoint is determined by the discrete values of the adjoint IND scheme and discretization and differentiation commute in the developed framework. iii) a proof that the values of the adjoint IND scheme corresponding to a BDF method with constant order and stepsize converge to the solution of the adjoint IVP on the open interval. In addition, a proof is given that demonstrates the convergence of the FE approximation to the weak solution of the adjoint IVP on the entire interval. iv) goal-oriented global error estimators for BDF methods that weight, for each integration step, a local error quantity with the corresponding value of the adjoint IND scheme and yields in sum an accurate and efficient estimate for the actual error. As local error quantity defect integrals and local truncation errors are employed, respectively. v) strategies for goal-oriented global error control in BDF methods that either adapt the locally acting relative tolerance or the given integration scheme using the stepwise error indicators. vi) an ODE model of an exothermic, self-accelerating chemical reaction with mass transfer carried out in a discontinuous Stirred Tank Reactor. With this real-world example from chemical engineering the applicability and reliability of the novel techniques for the approximation of weak adjoints and for the simulation with goal-oriented global error control are shown

On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies

Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors

In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step.Comment: 28 pages, 6 figure

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

On the smoothness of nonlinear system identification

We shed new light on the \textit{smoothness} of optimization problems arising in prediction error parameter estimation of linear and nonlinear systems. We show that for regions of the parameter space where the model is not contractive, the Lipschitz constant and $\beta$-smoothness of the objective function might blow up exponentially with the simulation length, making it hard to numerically find minima within those regions or, even, to escape from them. In addition to providing theoretical understanding of this problem, this paper also proposes the use of multiple shooting as a viable solution. The proposed method minimizes the error between a prediction model and the observed values. Rather than running the prediction model over the entire dataset, multiple shooting splits the data into smaller subsets and runs the prediction model over each subset, making the simulation length a design parameter and making it possible to solve problems that would be infeasible using a standard approach. The equivalence to the original problem is obtained by including constraints in the optimization. The new method is illustrated by estimating the parameters of nonlinear systems with chaotic or unstable behavior, as well as neural networks. We also present a comparative analysis of the proposed method with multi-step-ahead prediction error minimization

Strict bounding of quantities of interest in computations based on domain decomposition

This paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods. The proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems : on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver. Beside practical considerations on the parallel computation of the bounds, it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation. Academic assessments are given on 2D static linear mechanic problems.Comment: Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, online previe