Aggregation Dynamics: Numerical Approximations, Inverse Problems, and Generalized Sensitivity
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 post-gelation data.