  The use of ensemble methods to solve inverse problems is attractive because
it is a derivative-free methodology which is also well-adapted to
parallelization. In its basic iterative form the method produces an ensemble of
solutions which lie in the linear span of the initial ensemble. Choice of the
parameterization of the unknown field is thus a key component of the success of
the method. We demonstrate how both geometric ideas and hierarchical ideas can
be used to design effective parameterizations for a number of applied inverse
problems arising in electrical impedance tomography, groundwater flow and
source inversion. In particular we show how geometric ideas, including the
level set method, can be used to reconstruct piecewise continuous fields, and
we show how hierarchical methods can be used to learn key parameters in
continuous fields, such as length-scales, resulting in improved
reconstructions. Geometric and hierarchical ideas are combined in the level set
method to find piecewise constant reconstructions with interfaces of unknown
topology.

Numerical Analysis;Optimization and Control;Methodology