Joint Mathematics Colloquium
University of Idaho
|Inverse problems arise when we combine
measurements with mathematical models, and they occur in many areas of
science and engineering such as: Earth Sciences, Medical Imaging,
Electromagnetics and Epidemiology. An important consideration in
inverse problems is the fact that both measurements and models contain
errors or uncertainties. For example, there could be false
measurement readings, the model may be simplified, or solutions of the
model may be computationally approximated.
Most inverse problems are ill-posed because the underlying process we are trying to understand cannot be resolved by measurements that may be limited or contain inconsistent information. The problem can be made well-posed by adding information to it via regularization, and in this talk I will emphasize the statistical aspects of regularization. In particular, once we add regularization terms to the inverse problem we must appropriately weight information, and I will describe how to weight information according to its uncertainty. In addition, I will show how the statistical view of inverse problems gives new approaches to regularization and estimating uncertainty or weights on measurements and model. These approaches will be demonstrated on problems in Hydrology and Geophysics.