Efficient Inverse UQ for PDEs with Uncertain Data

Cinsider modelling a physical phenomena, where certain quantities or properties within the model are uncertain. For example modelling viscous fluid flow using the Navier-Stokes equations, where the fluid viscosity v is unknown. Or in modelling heat diffusion, where the thermal conductivity A of the material in question is unknown. Rather than assume a certain values for these quantities, it is oftern more insightful to model tham as random quantities (i.e. as functions of random variables (RV's). This leads to models described by PDEs with uncertain data, where that data may be a co-efficient, a source/forcing term or possibly on the demain in which it is to be solved. Solving these equations naturally leads to solutions which are themselves RVs. "Uncertainty quantification (UQ) is the tudy of the propagation of randomness in the data into randomness in some quantity of interest", for example the mean and variance of the solution.

As explained by Stuart, inverse problems are concerned with the (related) problem of determining an input u, from some noisy observations of data y. That is,

y = G(u) + n

where G is an observation operator (representing the fact that we can not measure y directly) and n represents the noise. It is often assumed that we have information about that noise, for example we may have n ~ N (0,E). Inverse problems of this kind centre around the idea of characterising the posterior possibility distribution. P (u|y)
by using Bayes' law;
P(A|B) x P(B|A)P(A)
for events A,B and prior beliefs on the prior distribution P(U)

This PhD project is concerned with both the forward problem and inverse problem methods of unvertainty quantification with the aim of bringing together techniques from each. It is hoped that studying both approaches will provide greater insight into the area as a whole, resulting in the creation of new efficient algorithms/methods for IQ. To begin with, techniques such as Stochastic Galerkin Finite Element Methods (Forward UQ) and Markov Chain Monte Carlo methods (Inverse UQ) will be studied before being applied to problems including, but not restricted to, those in areas of interest to the National Physical Laboratory (NPL)