  In this article we consider a Bayesian inverse problem associated to elliptic
partial differential equations (PDEs) in two and three dimensions. This class
of inverse problems is important in applications such as hydrology, but the
complexity of the link function between unknown field and measurements can make
it difficult to draw inference from the associated posterior. We prove that for
this inverse problem a basic SMC method has a Monte Carlo rate of convergence
with constants which are independent of the dimension of the discretization of
the problem; indeed convergence of the SMC method is established in a function
space setting. We also develop an enhancement of the sequential Monte Carlo
(SMC) methods for inverse problems which were introduced in \cite{kantas}; the
enhancement is designed to deal with the additional complexity of this elliptic
inverse problem. The efficacy of the methodology, and its desirable theoretical
properties, are demonstrated on numerical examples in both two and three
dimensions.
