UC San Diego

At their best, mathematical models of physical and biological systems strive to represent the nature faithfully while maximizing their simplicity and efficiency. Large numbers of variables involved in biophysical and geophysical processes make it necessary to develop predictive models with the minimum conceivable level of complexity. The combination of coupled phenomena, spatial and temporal scales, different regimes of behavior, and plethora of agents interacting simultaneously constitutes an unmanageable amalgamation of factors that unavoidably reduce both performance and tractability of mathematical models. Creating models with the optimal complexity requires finding the balance between the following characteristics: accuracy, computational cost, applicability to multiple scales and regimes, ability to represent realistic scenarios, and versatility. In this dissertation, we present analytical models of heat transfer in fractured porous media, in vitro and in vivo kinetics models of polymerization, and a hybrid algorithm for reaction-diffusion systems. These models provide more accurate representations of reality than their current counterparts, and they do so at the small fraction of the computational cost.