January 17, 2013

Speaker: Oksana Chkrebtii

Title: Probabilistic Solution of Differential Equations for Bayesian Uncertainty Quantification and Inference

Abstract: In many scientific disciplines the time and space evolution of system states is naturally described by differential equation models, which define states implicitly as functions of their own rates of change. Inference for differential equation models requires an explicit representation of the states (the solution) that is typically not known in closed form, but can be approximated by a variety of discretization-based numerical methods. However, the associated numerical error analysis cannot currently be propagated through the statistical inverse problem, and is thus ignored in practice.

We resolve this problem by developing a Bayesian framework to characterize discretization uncertainty in models defined by high-dimensional systems of differential equations with unknown solutions. Viewing solution estimation as an inference problem allows us to quantify and propagate discretization error into uncertainty in the model parameters and subsequent predictions. Our approach provides a statistical alternative to deterministic numerical integration techniques for estimation of complex dynamic systems, particularly those that are chaotic, ill-conditioned, or contain unmodelled functional variability. We discuss properties, efficient computational algorithms, and application to a wide range of challenging high-dimensional forward and inverse problems.