  Particle methods are widely used because they can provide accurate
descriptions of evolving measures. Recently it has become clear that by
stepping outside the Monte Carlo paradigm these methods can be of higher order
with effective and transparent error bounds. A weakness of particle methods
(particularly in the higher order case) is the tendency for the number of
particles to explode if the process is iterated and accuracy preserved. In this
paper we identify a new approach that allows dynamic recombination in such
methods and retains the high order accuracy by simplifying the support of the
intermediate measures used in the iteration. We describe an algorithm that can
be used to simplify the support of a discrete measure and give an application
to the cubature on Wiener space method developed by Lyons and Victoir [Proc. R.
Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198].
