Using the one-dimensional diffusion equation as an example, this seminar looks at ways of constructing approximations to the solution and coefficient functions of differential equations when the coefficients are not fully defined. There may, however, be some information about the solution. The input data, usually given as values of a small number of functionals of the coefficients and the solution, is insufficient for specifying a well-posed problem, and so various extra assumptions are needed. It is argued that looking at these inverse problems as problems in Bayesian statistics is a unifying approach. We show how the standard methods of Tikhonov Regularisation are related to special forms of random field. The numerical approximation of stochastic partial differential Langevin equations to sample generation will be discussed.
