A Measure Theoretic Computational Approach for Inverse Sensitivity Problems

ABstract.
We consider the inverse sensitivity analysis of a map from a set of
parameters and data to a quantity of interest. We are particularly
interested in implicitly-defined maps, e.g. involving the solution of
a differential equation. The inverse problem is to describe the random
variation in the input that leads to an imposed or observed random
variation in the output quantity. We formulate this as an ill-posed
inverse problem for an integral equation using the Law of Total
Probability. We then describe a computational method for computing
solutions that has two stages. In the first part, we approximate the
unique set-valued solution to the inverse of the integral equation
using derivative information. In the second part, we apply basic ideas
from measure theory to compute the approximate probability measure on
the parameter and data space that solves the integral equation. We
discuss convergence of the method, and explain how to use the method
to compute the probability of events in the input (parameter) space.
The talk is illustrated with a number of examples. We also discuss
briefly the numerical analysis (accuracy) of the method and the
consideration of multiple quantities of interest and data
assimilation.