Adaptive time-stepping methods for SPDEs and Multilevel Monte Carlo techniques for inverse prob

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The MCMC and HMC methods are well known methods for Bayesian inference to obtain parameters from data. Recent advances have examined multi-level version of these methods. There are a number of areas we would like to explore in the thesis: Develop novel time-stepping methods for SPDEs with non-globally Lipschitz drift with a view to extending to space-time adaptivity.At the moment a global refinement strategy is mostly used for MLMC. One area of interest is to combine the multilevel technique with a posteriori error techniques - where a FE mesh is refined to minimize the error of a given quantity of interest or use space-time adaptivity for mesh refinement. We plan to extend the work to examine model error in the approximation and also develop the proxy models. A number of numerical integration techniques are used - we have ideas around the choice of integrator that should improve the sampling and hence convergence of the MCMC chain.