Computational Mathematical Modeling: An Integrated Approach Across Scales

A significant challenge in the world of mathematical modeling is to develop a model or a series of models that can provide transitions from one spatial or temporal regime to another. For instance, models that connect molecular processes in cells to physiological function in organs, or map electron configurations to macroscopic properties of materials can be very useful. The current book aims to provide an approach to modeling that is integrated across scales and tuned in to computational issues.

The authors note that the nature of a problem can often suggest the mathematical tools appropriate to model it. So differential equations are appropriate in some instances (ranging from planetary motion to biochemical reactions), whereas stochastic modeling may be the best choice in other circumstances (in environments that are intrinsically probabilistic or in deterministic systems with significant random noise components). Accordingly, the book includes four chapters on deterministic models (mostly involving differential equations) and five chapters that involve some aspect of stochastic modeling.

Most of the deterministic models described in the book are compartment models. A compartment is an “aggregate which may increase in response to an input and decrease in response to an output.” The most interesting and useful compartment models are ones that connect multiple compartments. (Interacting population models fall into this category.) It is then sometimes possible to pass from discrete to continuous models by progressively refining or partitioning individual compartment models.

A chapter on dimensional analysis and scaling follows. It includes a treatment of the linear algebra of dimensional analysis and Buckingham’s Π-theorem on the null space of a system of dimensions. The topics treated here are explored more comprehensively than in comparable texts, and there are several good examples.

The treatment of stochastic modeling is focused on four areas: the modeling of noise, waiting processes, Markov processes, and cellular automata. Modeling with waiting time processes gets the longest discussion and is the best developed. It has three good examples: neuron firing and biological noise, stochastics in chemical kinetics, and photon migration with Levy flight.

This book developed from a one semester graduate course in mathematical modeling and would be appropriate for advanced undergraduates or graduate students in mathematics, engineering or the sciences.

While the book has some attractive individual features, it doesn’t really fit together in a satisfying way. It has the feel of an assembly of fragments without any real coherence, so the word “integrated” in the title is something of a puzzle. 

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.