Efficient Sampling of Dynamical Systems with Spatial Uncertainty

We present two methods for efficiently sampling the response (trajectory space) of dynamical systems operating under spatial uncertainty assumed to be representable with Gaussian processes. The dynamics of such systems depends on spatially indexed uncertain parameters that span infinite dimensional spaces. This places a heavy computational burden on the implementation of existing methodologies, a challenge addressed with two new conditional sampling approaches. When a single instance of the uncertainty is needed in the entire domain, we use a fast Fourier transform technique. When the Gaussian process has a compactly supported kernel, we use an incremental sampling approach, which not only is fast but also has a very small memory footprint. We prove that both methods produce the same distributions as the widely used Choleskybased approaches while having much less complexity. We illustrate this convergence at a far smaller computational effort and memory cost for a simple vehicle model.