Department of Mathematics Colloquium
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Abstract |
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| 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.
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