  We consider a randomised implementation of the finite element method (FEM)
for elliptic partial differential equations on high-dimensional models. This is
motivated by applications where model predictions are essential for real-time
process diagnostics. In these circumstances it is imperative to expedite
prediction without a significant compromise in the model's fidelity, which in
turn relies on the rapid assembly and solution of the associated system of
equations typically at the many-query context. Our approach involves converting
the solution of the linear, symmetric positive definite FEM system into an
over-determined least squares problem, whose solution is then projected onto a
low-dimensional subspace. The resulting low-dimensional system can be
effectively sketched as a product of two high-dimensional matrices using a
parameter-dependent non-uniform sampling distribution, utilising only a small
subset of the model's parameters. Although different to the optimal sampling
distributions based on the statistical leverage-scores of the rows of the
matrices, we show that the distance between them shrinks for an appropriate
choice of the projection subspace. For the approximate solution we bound the
incurring errors due to the projection, subspace approximation and sketching
and show that the overall error is dominated by the condition number of the
projected stiffness matrix. Our approach is tested on simulations on the
Dirichlet and Neumann problems for the steady-state diffusion equation. The
results show that our approach has on average a tenfold improvement on the
computational times compared to the classical deterministic framework at the
expense of a moderately small approximation error.
