The effect of problem perturbations on nonlinear dynamical systems and their reduced order models

Reduced order models are used extensively in many areas of science and engineering for simulation, design, and control. Reduction techniques for nonlinear dynamical systems produce models that depend strongly on the nominal set of parameters for which the reduction is carried out. In this paper we address the following two questions: 'What is the effect of perturbations in the problem parameters on the output functional of a nonlinear dynamical system?' and 'To what extent does the reduced order model capture this effect?

Error Estimation for Reduced Order Models of Dynamical Systems

The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude

Digital Filter Stepsize Control of DASPK and its Effect on Control Optimization Performance, M.Sc. Thesis

It has long been known that the solutions produced by adaptive solvers for ordinary differential (ODE) and differential algebraic (DAE) systems, while generally reliable, are not smooth with respect to perturbations in initial conditions or other problem parameters. Söderlind and Wang [12, 13] have recently developed a digital filter stepsize controller that has a theoretical basis from control and appears to result in a much smoother dependence of the solution on problem parameters. This property seems particularly important in the control and optimization of dynamical systems, where the optimizer is generally expecting the DAE solver to return solutions that vary smoothly with respect to the parameters. We have implemented the digital filter stepsize controller in the DAE solver DASP K3.1, and used the new solver for the optimization of dynamical systems. The improved performance of the optimizer, as a result of the new stepsize controller, is demonstrated on a biological problem regarding the heat shock response of Escherichia coli. ∗This work was supported by DOE DE-FG03-00ER25430, NSF/ITR ACI-0086061, NSF CTS

Model order reduction for nonlinear IC models

Model order reduction is a mathematical technique to transform nonlinear dynamical models into smaller ones, that are easier to analyze. In this paper we demonstrate how model order reduction can be applied to nonlinear electronic circuits. First we give an introduction to this important topic. For linear time-invariant systems there exist already some well-known techniques, like Truncated Balanced Realization. Afterwards we deal with some typical problems for model order reduction of electronic circuits. Because electronic circuits are highly nonlinear, it is impossible to use the methods for linear systems directly. Three reduction methods, which are suitable for nonlinear differential algebraic equation systems are summarized, the Trajectory piecewise Linear approach, Empirical Balanced Truncation, and the Proper Orthogonal Decomposition. The last two methods have the Galerkin projection in common. Because Galerkin projection does not decrease the evaluation costs of a reduced model, some interpolation techniques are discussed (Missing Point Estimation, and Adapted POD). Finally we show an application of model order reduction to a nonlinear academic model of a diode chain