  As stakeholders and policy makers increasingly rely upon quantitative
predictions from advanced computational models, a problem of fundamental
importance is the quantification and reduction of uncertainties in both model
inputs and output data. The typical work-flow in the end-to-end quantification
of uncertainties requires first formulating and solving stochastic inverse
problems (SIPs) using output data on available quantities of interest (QoI).
The solution to a SIP is often written in terms of a probability measure, or
density, on the space of model inputs. Then, we can formulate and solve a
stochastic forward problem (SFP) where the uncertainty on model inputs is
propagated through the model to make quantitative predictions on either
unobservable or future QoI data.
  In this work, we use a measure-theoretic framework to formulate and solve
both SIPs and SFPs. From this perspective, we quantify the geometric
characteristics of using hypothetical sets of QoI that describe both the
precision and accuracy in solutions to both the SIP and SFP. This leads to a
natural definition of the optimal experimental design, i.e., what the optimal
configuration a finite set of sensors is in space-time. Several numerical
examples and applications are discussed including the ADvanced CIRCulation
(ADCIRC) model using simulated data from Hurricane Gustav to determine an
optimal placement of buoys in the Gulf of Mexico to capture high water marks of
the storm surge in order to reduce uncertainties in bottom roughness
characteristics.
