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Error-controlled global sensitivity analysis of ordinary differential equations. (English) Zbl 1252.65017
Summary: We propose a novel strategy for global sensitivity analysis of ordinary differential equations. It is based on an error-controlled solution of the partial differential equation (PDE) that describes the evolution of the probability density function associated with the input uncertainty/variability. The density yields a more accurate estimate of the output uncertainty/variability, where not only some observables (such as mean and variance), but, also, structural properties (e.g., skewness, heavy tails, bi-modality) can be resolved up to a selected accuracy.
For the adaptive solution of the PDE Cauchy problem, we use the Rothe method with multiplicative error correction, which was originally developed for the solution of parabolic PDEs. We show that, unlike in parabolic problems, conservation properties necessitate a coupling of temporal and spatial accuracy to avoid accumulation of spatial approximation errors over time. We provide convergence conditions for the numerical scheme and suggest an implementation using approximate approximations for spatial discretization to efficiently resolve the coupling of temporal and spatial accuracy. The performance of the method is studied by means of low-dimensional case studies. The favorable properties of the spatial discretization technique suggest that this may be the starting point for an error-controlled sensitivity analysis in higher dimensions.
MSC:
65C30 Numerical solutions to stochastic differential and integral equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Software:
Matlab
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