Daniel Wirtz

A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems

In this work an efficient approach for a posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time- and parameter-affine linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the systems nonlinearity by the DEIM method [S. Chaturantabut and D. C. Sorensen, SIAM J. Sci. Comput., 32 (2010), pp. 2737--2764]. The proposed a posteriori error estimator can be efficiently decomposed in an offline/online fashion and is obtained by a one-dimensional auxiliary ODE during reduced simulations. Key elements for efficient online computation are partial similarity transformations and matrix-DEIM approximations of the nonlinearity Jacobians. The theoretical results are illustrated by application to an unsteady Burgers equation and a cell apoptosis model.

Efficient a-posteriori error estimation for nonlinear kernel-based reduced systems

Abstract In this paper, we consider the topic of model reduction for nonlinear dynamical systems based on kernel expansions. Our approach allows for a full offline/online decomposition and efficient online computation of the reduced model. In particular, we derive an a-posteriori state-space error estimator for the reduction error. A key ingredient is a local Lipschitz constant estimation that enables rigorous a-posteriori error estimation. The computation of the error estimator is realized by solving an auxiliary differential equation during online simulations. Estimation iterations can be performed that allow a balancing between estimation sharpness and computation time. Numerical experiments demonstrate the estimation improvement over different estimator versions and the rigor and effectiveness of the error bounds.

A-Posteriori Error Estimation for Parameterized Kernel-Based Systems

Abstract This work is concerned with derivation of fully offline/online decomposable efficient a-posteriori error estimators for reduced parameterized nonlinear kernel-based systems. The dynamical systems under consideration consist of a nonlinear, time- and parameter-dependent kernel expansion representing the systems inner dynamics as well as time- and parameter-affine inputs, initial conditions and outputs. The estimators are established for a reduction technique originally proposed in Phillips et al. (2003) and are an extension of the estimators derived in Wirtz and Haasdonk (2012) to the fully time-dependent, parameterized setting. Key features for the efficient error estimation are to use local Lipschitz constants provided by a certain class of kernels and an iterative scheme to balance computation cost against estimation sharpness. Together with the affinely time/parameter-dependent system components a full offline/online decomposition for both the reduction process and the error estimators is possible. Some experimental results for synthetic systems illustrate the efficient evaluation of the derived error estimators for different parameters.

The rocky road to extended simulation frameworks covering uncertainty, inversion, optimization and control ☆

Abstract In the past decades, simulation frameworks have greatly increased in complexity, due to coupling of models from various disciplines into so-called integrated models. Recently, the combination with tools for uncertainty quantification, inverse modelling, optimization and control started a development towards what we call extended simulation frameworks. While there is an ongoing discussion on quality assurance and reproducibility for simulation frameworks, we have not observed a similar discussion for the extended case. Particularly for extended frameworks, the need for quality assurance is high: The overwhelming range of options and algorithms is unmanageable by a domain expert and opaque to decision makers or the public. The resulting demand for ‘intelligent software’ with automated configuration can lead to a blind trust in simulation results even if they are incorrect. This is a threatening scenario due to potential consequences in simulation-based engineering or political decisions. In this paper, we analyze the increasing complexity of scientific computing workflows, and discuss the corresponding problems of extended scientific simulation frameworks. We propose a paradigm that regulates the allowable properties of framework components, supports the framework configuration for complex simulations, enforces automatic self-tests of configured frameworks, and communicates automated algorithm choices, potentially critical user settings or convergence issues with adaptive detail level and urgency to the end-user. Our goal is to start transferring the quality assurance discussion in the field of integrated modeling and conventional software frameworks to the area of extended simulation frameworks. With this, we hope to increase the reliability and transparency of (extended) frameworks, framework use and of the corresponding simulation results.