A Hyperspherical Method for Discontinuity Location in Uncertainty Quantification

When an uncertain system is modeled statistically, we may want to detect the
chances that some quantity of interest Q exceeds a tolerance value.  We may be
looking for severe rain events, excessive heat buildup, or surges in
electrical power demand.

The probabilistic parameter spaces may be smooth, but the region of
extreme Q may be disconnected, irregular in shape, or relatively small.
We may think of the boundary between the regions of acceptable and extreme Q
as a kind of discontinuity.  When the uncertainty is modeled by a large
number of independent variables, it becomes difficult to determine the 
boundary of this region, or sample its interior.  Monte Carlo methods and
hierarchical adaptive sparse grids are frequently used in this case. 

An alternative method is proposed, involving a mapping onto a hypersphere,
which allows us to unfold the discontinuity surface and make good estimates 
of the shape and probabilistic volume.