Extension of the Computational Singular Perturbation Method to System of Stiff Differential Equations with Uncertain Parameters

Modelling complex dynamical system often involves solving systems of ordinary differential equations (ODE). The time-scales in such systems can span orders of magnitude and such stiff problems present stability issues and severe time-step restrictions for their time-integration using explicit methods. The Computational Singular Perturbation method has been found to be an effective tool to identify and decouple different disparate time-scales, allowing reduction of model complexity and stiffness, through a separation of the dynamics into slow and fast ones for which different integration schemes can be employed. In this talk, we discuss the extension of the CSP method to problems involving uncertain parameters and data that are considered as random variables with known probability laws.

The solution to the ODE system is then random and sought in a finite dimensional stochastic subspace constructed by mean of orthogonal functionals in the random parameters (e.g. Polynomial Chaos bases). We discuss the determination of the slow and fast stochastic subspaces of the dynamics, which are spanned by the eigenvectors of the Jacobian of the stochastic dynamics. This lead us to the resolution of a stochastic eigenproblem (reformulated as a set of nonlinear problems by means of a Galerkin projection on the stochastic basis) and a redefinition of the so-called exhaustion criterion to partition the eigenmodes into slow and fast sets. Finally, we propose a simplified approach based on the projection of the dynamics onto deterministic subspaces, for the case where the stochastic eigenvectors are weakly affected by the uncertainty. This simplified approach alleviate the computation of the stochastic eigenvectors with significant computational savings.
The two stochastic CSP methods will be tested on a simple system with different level of uncertainty to assess their accuracy and efficiency.