On Optimal Sampling Designs to Estimate Unknown Parameters
in Elliptic Partial Differential Equations
Nathanial Burch, M.S. Candidate, Department of Statistics, Colorado State University.
Wednesday, April 1, 2009
1:15 pm, Wagar 132
We consider the inverse problem of estimating unknown parameters in elliptic partial differential equations given a small number of measurements. Such measurements are each prone to probabilistic measurement errors with known distribution. In fairly simple cases, the estimates are obtained via least-squares methods, whereas a Bayesian framework is discussed for more complicated problems. We point out, and illustrate with several examples, that the sampling design, i.e., the locations of the measurements, can have a large impact on the resulting estimates. We explain this behavior through a generalized leverage measure and through sensitivities of the solution with respect to the unknown parameters. Using just these two quantities, which can be computed a priori, we describe an efficient sampling design to minimize the mean squared error of the estimates. That is, given a small number of sensors (limited by cost, time, physical constraints, etc.), we address the important design question of the optimal placement of the sensors. Several extensions and adaptations of this research are also explored, all from the point of view of attempting to design an optimal sensing strategy.
Dr. Jennifer Hoeting (Advisor)
Dr. Donald Estep (Co-advisor)
Dr. Edwin Chong (Electrical and Computer Engineering, Outside Member)