Partial Differential Equations (PDEs) are a central concept in scientific simulation. A large class of contemporary PDE solvers fits into the framework of iteratively linearized least-squares estimation, and is thus closely related to the Gaussian process regression framework foundational to Bayesian machine learning. In the PhD project, Tim will use this connection to develop practically useful probabilistic PDE solvers — i.e. simulation methods for PDE problems that return a probability measure instead of a point estimate. Practical considerations like computational efficiency, efficient datastructures, and empirical calibration of the uncertainty estimate will be central goals. The visit to Aalto will provide an opportunity to also connect the project to related ideas from signal processing, and to explore opportunities for massive parallelization.