Mathematics Colloquium: Inverse Problems and Uncertainty Quantification

Combining physical or mathematical models with observational data often results in an ill-posed inverse problem. Regularization is typically used to solve ill-posed problems, and it can be viewed as adding statistical or probability information to the problem in the form of uncertainties. These uncertainties occur in the model, parameters or measurements and quantifying them is a challenge. The Bayesian interpretation of uncertainties formalizes the process of updating prior beliefs with observational data, however, it can be computationally demanding. Alternatively, we develop one of the simplest forms of regularization, least squares, to estimate prior uncertainty and call it the chi-squared method. The chi-squared method is compared to Bayesian inference on an event reconstruction problem and we find it to be slightly less accurate but more efficient. In the hope of getting more accurate results, we further develop the chi-squared method to get more dense estimates of prior uncertainty. We then use it to estimate soil moisture in the Dry Creek Watershed near Boise, Idaho. Finally, the chi-squared method will be extended to nonlinear problems, and results from cross-well tomography and soil electrical conductivity will be shown.