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

MCMC和HMC方法是用于贝叶斯推断以从数据获得参数的公知方法。最近的进展已经检查了这些方法的多层次版本。本论文主要研究的内容有：针对非全局Lipschitz漂移的SPDE问题，发展新的时间步长方法，并将其扩展到时空自适应问题。感兴趣的一个领域是结合联合收割机的多级技术与后验误差技术-其中有限元网格细化，以最大限度地减少一个给定的感兴趣的量的误差或使用空间-时间自适应网格细化。我们计划扩展工作，以检查近似模型的误差，并开发代理模型。一些数值积分技术的使用-我们有想法的积分器的选择，应改善采样，因此收敛的MCMC链。

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.