Global Sensitivity Analysis of Ordinary Differential Equations

Abstract

Ordinary differential equations play an important role in the modeling of many real-world processes. To guarantee reliable results, model design and analysis must account for uncertainty and/or variability in the model input. The propagation of uncertainty & variability through the model dynamics and their effect on the output is studied by sensitivity analysis. Global sensitivity analysis is concerned with variations in the model input that possibly span a large domain. Two major problems that complicate the analysis are high-dimensionality and quality control, i.e. controlling the approximation error of the estimated output uncertainty. Current numerical approaches to global sensitivity analysis mainly focus on scalability to high-dimensional models. However, to what extent the estimated output uncertainty approximates the true output uncertainty generally remains unclear. In this thesis we suggest an error-controlled approach to global sensitivity analysis of ordinary differential equations. The approach exploits an equivalent formulation of the problem as a partial differential equation, which describes the evolution of the state uncertainty in terms of a probability density function. We combine recent advances from numerical analysis and approximation theory to solve this partial differential equation. The method automatically controls the approximation error by adapting both temporal and spatial discretization of the numerical solution. Error control is realized using a Rothe method that provides a framework for estimating temporal and spatial errors such that the discretization can be adapted accordingly. We use a novel technique called approximate approximations for the spatial discretization; it is the first time that these are used in the context of an adaptive Rothe scheme. We analyze the convergence of the method and investigate the performance of approximate approximations in the adaptive scheme. The method is shown to converge, and the theoretical results directly indicate how to design an efficient implementation. Numerical examples illustrate the theoretical results and show that the method yields highly accurate estimates of the true output uncertainty. Furthermore, approximate approximations have favorable properties in terms of readily available error estimates and high approximation order at feasible computational costs. Recent advances in the theory of approximate approximations, based on a meshfree discretization of the state space, promise that the applicability of the adaptive density propagation framework developed herein can be extended to higher-dimensional problems.