MCMC for Bayesian Inverse Problems in PDEs

James Rynn (The University of Manchester)

ATB Frank Adams 1,

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.

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