  In this paper we tackle the problem of generating uniform designs in very
small subregions of computer model input space that have been identified in
previous experiments as worthy of further study. The method is capable of
producing uniform designs in subregions of computer model input space defined
by a membership function that consists of a continuous function passing a
threshold test, and does so far more efficiently than current methods when
these subregions are small. Our application is designing for regions of input
space that are not ruled out by history matching, a statistical methodology
applied in numerous diverse scientific applications whereby model runs are used
to cut out regions of input space that are incompatible with real world
observations. History matching defines a membership function for a region of
input space that is not ruled out yet by observations in the form of a distance
metric called implausibility. We use this distance metric to drive a new type
of Evolutionary Monte Carlo algorithm with a uniform distribution on the not
ruled out yet region as its target distribution. The algorithm can locate and
generate uniform points within extremely small subspaces of the computer model
input space with complex and even disconnected topologies. We illustrate the
performance of the technique in comparison to current methods with a number of
idealised examples. We then apply our algorithm to generating an optimal design
for the not ruled out yet region of a galaxy simulation model called GALFORM
following 4 previous waves of history matching where the target region is
0.001% the volume of the input space.
