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.

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