## MCMC for Bayesian Inverse Problems in PDEs

#### James Rynn (The University of Manchester)

Suppose you have noisy observations of a system determined by a parameter $$\theta$$,
$\mathbf{D} = \boldsymbol{\mathcal{G}}(\theta) + \boldsymbol{\eta}, \quad \quad \quad \boldsymbol{\eta} \sim \mathcal{N}(\boldsymbol{0}, \Sigma),$
here the function $$\boldsymbol{\mathcal{G}}(\theta)$$ maps parameters onto observable space.
We consider the case where $$\boldsymbol{\mathcal{G}}(\theta)$$ can be approximated by the solution of a PDE, with input data (such as coefficients, initial or boundary conditions) described by the parameter $$\theta$$.
Posing this as a Bayesian inverse problem, we look at how Markov chain Monte Carlo techniques may be used to quantify uncertainty about the parameter $$\theta$$ and hence the system as a whole.