A computational measure theoretic approach to inverse sensitivity problems |
Don Estep, PhD, Department of Statistics, Colorado State University Mathematics and Statistics.
Monday, April 27, 2009
4:oo pm, 223 Weber
ABSTRACT |
We consider the inverse problem for a finite dimensional map determined implicitly by the solution of a nonlinear system of equations. An example is the map from the space of parameters for a differential equation to a functional computed from the solution of the equation. We focus on problems where the uncertainty in the quantity of interest is represented by a random variable with a given distribution. We then seek to describe the corresponding random uncertainty in the inputs.
We describe an efficient computational measure-theoretic method for this probabilistic inverse problem. The method has two stages. First, we approximate the unique inverse of the many-to-one map into a set of generalized contours using the adjoint operator to define approximate generalized contours. We then derive an efficient computational measure theoretic approach to use the inverse on the set of approximate contours to compute an approximate probability measure on the input space using a simple function approximation to the posterior density function. We discuss convergence of the method, and explain how to use the method to solve goal oriented inverse problems such as computing the probability of events in the input (parameter) space and application to Bayesian analysis.