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Academic Lectures
AMS Colloquium Friday 2/21/25 at 3pm Chauvenet Hall 143
Department / Organization: AMS
Dr. Wolfgang Bangerth, CSU - On the notion of "information" in inverse problems
Abstract Inverse problems are ones where one would like to reconstruct a spatially variable function from measurements of a system in which this function appears as a coefficient or right hand side. Examples are biomedical imaging and seismic imaging of the earth.
In many inverse problems, practitioners have an intuitive notion of how much one "knows" about the coefficient in different parts of the domain - that is, that there is a spatially variable "amount of information". For example, in seismic imaging, we can only know about those parts of the earth that are traversed by seismic waves on their way from earthquake to receiving seismometer, whereas we cannot know about places not on these raypaths.
Despite the fact that this concept of "information" is intuitive to understand, there are no accepted quantitative measures of information in inverse problems. I will present an approach to define such a spatially variable "information density" and illustrate both how it can be computed practically, as well as used in applications. The approach is based on techniques borrowed from Bayesian inverse problems, and an approximation of the covariance matrix using the Cramer-Rao bound.
