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 approximation.

PGD algorithms based on rigorously derived least-squares formulations are developed
and it is shown that convergence of associated greedy algorithms is guaranteed.
These formulations also have the added benefit that they remove the requirement
to satisfy stability conditions related to weakly coercive problems. A variety
of least-squares formulations are derived based on different first-order
reformulations of the problems and a thorough comparison is made. The
least-squares PGD algorithms developed in this research are applied once again to the Poisson and
Stokes problems as well as the non-symmetric convection-diffusion equation.

Finally, an application of the PGD to a deterministic approach to kinetic theory models
in polymer rheology is considered. This involves solving the (potentially
high-dimensional) Fokker-Planck equation. Results are provided for a spatially
homogeneous form of the Fokker-Planck equation and streamline upwinding is employed to stabilise
the numerical solutions. A method recently proposed for solving the fully
non-homogeneous Fokker-Planck equation is investigated which uses an operator
splitting technique. It is shown that this approach is not suitable to be
applied in conjunction with the PGD and instead two different schemes for
solving this problem are proposed.