Mathematical models based on partial differential equations (PDEs) play a central role in science and engineering. This thesis aims to present some new perspectives on two different topics regarding such models.

The first issue is concerned with how mismatch between the model and the real system can often manifest itself. One way of dealing with such mismatch is to introduce stochasticity to model any uncertainty present. Concurrently, the increasing prevalence of instrumentation within modern engineering systems has driven the need for tools to combine sensor data with model predictions to help mitigate any model misspecification. This has led to an increased interest in uncertainty quantification (UQ). In the first part of this thesis we provide a review of the statistical finite element method (StatFEM) which is an emerging probabilistic method that allows observations of a physical system to be synthesised with the numerical solution of a PDE. We present a new theoretical analysis of StatFEM demonstrating that it has similar convergence properties to the finite element method on which it is based. Numerical examples are presented to demonstrate our theory, including an example which tests the robustness of StatFEM when extended to nonlinear quantities of interest.

The second issue is concerned with the fact that PDE models often involve a large number of parameters increasing the computational burden. Thus, the study of methods to perform dimension reduction are of vital importance. In the second part of this thesis we motivate the idea of localised active subspaces for dimension reduction, where the dimension reducing subspace varies smoothly as a function of the input parameters. To estimate localised active subspaces we then present a fully Bayesian model based on Gaussian Processes, called GrassGP, which can be used to interpolate a field of subspace-valued response-variables. Numerical experiments are presented to demonstrate this.