Extra Colloquium Mathematics, Professor Mark Girolami

Title: Quantifying Epistemic Uncertainty in ODE and PDE Solutions using
        Gaussian Measures and Feynman-Kac Path Integrals

Abstract:

Diaconis and O'Hagan originally set out a programme of research suggesting the evaluation of a functional can be viewed as an inference problem. This perspective naturally leads to construction of a probability measure describing the epistemic uncertainty associated with the evaluation of functions solving for systems of Ordinary Differential Equations (ODE) or a Partial Differential Equation (PDE). By defining a joint Gaussian Measure on the Hilbert space of functions and their derivatives appearing in an ODE or PDE a stochastic process can be

constructed. Realisations of this process, conditional upon the ODE or PDE, can be sampled from the associated measure defining "Global" ODE/PDE solutions conditional on a discrete mesh. The sampled realisations are consistent estimates of the function satisfying the ODE or PDE system and the associated measure quantifies our uncertainty in these solutions given a specific discrete mesh.

Likewise an unbiased estimate of the "Local" solutions of certain classes of PDEs, along with the associated probability measure, can be obtained by appealing to the Feynman-Kac identities and 'Bayesian Quadrature' which has advantages over the construction of a Global solution for inverse problems. In this talk I will describe the quantification of uncertainty using the methodology above and illustrate with various examples of ODEs and PDEs in specific inverse problems.

Colloquium coordinators are Prof.dr. A.C.D. van Enter (e-mail : A.C.D.van.Enter@rug.nl) and
Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl )

http://www.rug.nl/research/jbi/news/colloquia/mathematics-colloquia/