Proper generalised decompositions: theory and applications

In this thesis a recently proposed method for the efficient approximation of solutions to high-dimensional partial differential equations has been investigated. This method, known as the Proper Generalised Decomposition (PGD), seeks a separated representation of the unknown field which leads to the solution of a series of low-dimensional problems instead of a single high-dimensional problem. This effectively bypasses the computational issue known as the `curse of dimensionality'. The PGD and its recent developments are reviewed and we present results for both the Poisson and Stokes problems. Furthermore, we investigate convergence of PGD algorithms by comparing them to greedy algorithms which have previously been studied in the non-linear approximation community. We highlight that convergence of PGD algorithms is not guaranteed when a Galerkin formulation of the problem is considered. Furthermore, it is shown that stability conditions related to weakly coercive problems (such as the Stokes problem) are not guaranteed to hold when employing a PGD approximatio