A Measure-Theoretic Computational Method for Inverse Sensitivity Problems III: Multiple Quantities of Interest

Abstract

We consider inverse problems for a deterministic model in which the dimension of the output quantities of interest computed from the model is smaller than the dimension of the input quantities for the model. In this case, the inverse problem admits set-valued solutions (equivalence classes of solutions). We devise a method for approximating a representation of the set-valued solutions in the parameter domain. We then consider a stochastic version of the inverse problem in which a probability distribution on the output quantities is specified. We construct a measure-theoretic formulation of the stochastic inverse problem, then develop the existence and structure of the solution using measure theory and the disintegration theorem. We also develop and analyze an approximate solution method for the stochastic inverse problem based on measure-theoretic techniques. We demonstrate the numerical implementation of the theory on a high-dimensional storm surge application where simulated noisy surge data from Hurricane Katrina is used to determine the spatially variable bathymetry fields of highest probability.