Aggregation Dynamics: Numerical Approximations, Inverse Problems, and Generalized Sensitivity
Lieutenant Colonel Dustin D. Keck
Applied Mathematics, University of Colorado Boulder
Date and time:
Wednesday, May 21, 2014 - 1:00pm
Location:
ECCR 257 - Newton Lab
Abstract:
In this dissertation, we investigate several important mathematical and
computational issues that arise when using the Smoluchowski coagulation
equation as a model for bacterial aggregation. In particular, we study
the accuracy and robustness of numerical simulations and their impact
upon related inverse problems. We also study how generalized sensitivity
enhances experimental design optimization with an ultimate goal of
comparing with experimental data.
First, we study the impact of discretization strategy on the accuracy of
solution moment. We perform this investigation in anticipation of comparing
with different distributions moments reported by specific experimental
devices. For multiplicative aggregation kernels, finite volume methods
are superior to finite element methods both in accuracy and computational
effort. Conversely, for slowly aggregating systems the finite element
approach can produce as little error as the finite volume approach
and achieves more accuracy approximating the zeroth moment (at a substantially
reduced computational cost).
A better understanding of bacterial aggregation dynamics could also
lead to improvements in the treatment of bacterially mediated, life-threatening
human illnesses. Therefore, to reach towards our ultimate goal, we examine
the inverse problem of estimating the aggregation rate from experimental
data. In this study, we develop a methodology for a software implementation
of parameter fitting when solving inverse problems involving
the Smoluchowski coagulation equation. Additionally, we make the
novel extension of generalized sensitivity functions (GSFs) for ordinary
differential equations to GSFs for partial differential equations. We analyze
the GSFs in the context of size-structured population models, and
specifically analyze the Smoluchowski coagulation equation in order to
determine the most relevant time and volume domains for three distinct
aggregation kernels. Finally, we provide evidence that parameter estimation
for the Smoluchowski coagulation equation does not require postgelation
data.