# Partial Differential Equations

## Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems

This paper contributes a Bayesian probabilistic numerical method for linear and nonlinear partial differential equations (PDEs) and studies incorporation of the posterior measure produced into Bayesian inversion problems, in a prototypical pipeline of computation.

The central contribution is a probabilistic numerical method for linear PDEs. It is shown that when the PDE is linear and the prior measure on the solution is Gaussian with an appropriate covariance function, the posterior measure produced is also Gaussian and has a closed form. Furthermore, the posterior mean of this measure is identical to the point estimate of the solution produced by symmetric collocation.  This allows a transfer of convergence results from that work. New results are also developed surrounding the concentration of the posterior measure as the fidelity of the approximation is increased.

The second contribution of the paper is to propagate the uncertainty from the forward solver into the posterior measure over the solution of PDE-constrained Bayesian inverse problems. It is shown that when the posterior uncertainty over the forward solution is marginalised in the likelihood of the Bayesian inverse problem, the resultant posterior measure for the inverse problem rigorously accounts for the uncertainty resulting from an intractable forward solution. This allows for a cheap, approximate solution to the PDE to be used, which can reduce the computational time required for these challenging problems. The approach is applied to Electrical Impedance Tomography, a PDE-constrained inverse problem in medical imaging.

Lastly, posterior measures are constructed for a certain class of semilinear PDEs, such as the steady-state Allen-Cahn equation, a PDE known to exhibit multiple solutions. While the method proposed is heavily dependent on the form of the PDE, it illustrates how PN methods might be applied to other challenging nonlinear problems for which discretisation error is highly significant.

Video: View a short introduction to probabilistic meshless methods.

### References

1. Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems. Cockayne, J., Oates, C. J., Sullivan, T., Girolami, M. Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering.
2. Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems. Cockayne, J., Oates, C. J., Sullivan, T., Girolami, M., arXiv:1605.07811.