Uncertainty Quantification in the Presence of Observations

Andrew Stuart, U Warwick

Abstract: There is a significant body of research work concerning the quantification of uncertainty in the solution of elliptic PDEs with random coefficients. Typically this work makes the assumption that the diffusion coefficient is a random field with a simple representation via a Karhunen-Loeve or polynomial chaos representation. However, in many applications of interest this (prior) probabilistic information must be conditioned on observational data, leading to a posterior probability measure which has a much more complicated structure. For example in the study of groundwater flow it is natural to condition the conductivity (diffusion coefficient) on noisy observations of the solution to the elliptic PDE for the pressure.

In this talk I will show how the development of Bayesian statistics on function space provides
a natural framework for the study of such problems. For illustrative pruposes I will concentrate on the problem of groundwater flow. I will develop prior probability meausures using KL expansions in wavelet or Fourier bases, and show how to
condition these measures on data. I will develop a theory of well-posedness for the inverse problem and show how this leads to stability of the posterior measure with respect to changes in data, and finite truncation of the KL expansion.

Joint work with Masoumeh Dashti and Stephen Harris.