Title: A Sample-Free Method to Approximating a Probability Measure for Inverse Problems

Speaker: Troy Butler, Colorado State University

Date/Time: Wednesday, February 18, 2009, 2:00 pm

Location: CSRI Building, Room 95 (Sandia NM)

Brief Abstract: We consider the inverse problem of quantifying the uncertainty of inputs to a finite dimensional map, e.g. determined implicitly by solution of a nonlinear system, given specified uncertainty in a linear functional of the output of the map. The uncertainty in the output functional might be suggested by experimental error or imposed as part of a sensitivity analysis. We describe this problem probabilistically, so that the uncertainty in the quantity of interest is represented by a random variable with a known distribution, and we assume that the map from the input space to the quantity of interest is smooth. We derive an efficient method for determining the unique solution to the problem of inverting through a many-to-one map by inverting into a quotient space representation of the input space which combines a forward sensitivity analysis with the Implicit Function Theorem. We then derive an efficient computational measure theoretic approach to further invert into the entire input space resulting in an approximate probability measure on the input space. We also consider the effect of various sources of error on the inverse problem. There is a statistical error due to finite sampling, and two sources of deterministic errors due to approximations of the quantity of interest map. We provide an a priori result for the error resulting from linearization, and a posteriori error estimates for both the numerical error of solutions used in the linearization of the map and the error in the distribution due to sampling.

CSRI POC: James Stewart, (505) 844-8630