Quantifying Uncertainty in the Solution of Differential Equations: A Probabilistic Framework

Mark Girolami (University of Warwick)

Friday 30th May, 2014 15:00-16:00 Maths 204


Solving the forward and inverse problems when quantifying uncertainty in models of physical systems described by ordinary and partial differential equations requires a coherent probabilistic framework. Quantifying sources of uncertainty in the forward problem must include the non-analytic nature of solutions of ordinary and partial differential equations which in all but the simplest cases demands finite dimensional functional approximations based on for example finite elements, and discrete time numerical integration. The epistemic nature of this uncertainty can be formally defined by imposing appropriate prior measures on the Hilbert space of vector fields and corresponding functional solutions of the systems of equations via the Radon-Nikodyn derivative leading to continuous posterior measures over solutions. This work describes such methods to probabilistically solve ordinary and partial differential equations with proofs of consistency provided. Examples will include uncertainty quantification for Navier-Stokes equations, chaotic partial differential equations (Kuramoto-Shivasinsky), and biochemical kinetics.

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