
Math 80770, Fall 2006
Multi scale modeling
MWF 1:55 pm  2:45 pm, Edward J. DeBartolo Hall 240
Instructor: Mark Alber (6318371),
malber@nd.edu
Syllabus:
Modeling and simulation are becoming central research tools in
biology. The most advanced of these efforts have focused on single
levels or scales, e.g., genomic/proteomic, cellular, tissue, organ,
whole body, behavioral, and population. One now needs to develop the
mathematical approaches and computational tools to integrate models
from microscales to macroscales in a seamless fashion. Such multi
scale approaches are essential for producing quantitative,
predictive models of complex biological behaviors such as embryonic
development, cancer, cytoskeletal function, and ecosystems. At the
same time, developing the abstractions to integrate between scales
will lead to a much deeper understanding of the universal or generic
features of biological phenomena.
Multi scale modeling is a rapidly developing scientific field that
spans many disciplines including physics, biology, chemistry,
mathematics, statistics, engineering, and materials science. The
main idea of this approach is straightforward: one computes
information at a smaller (finer) scale and passes it to a model at a
larger (coarser) scale by leaving out degrees of freedom as one
moves from finer to coarser scales. Within this context, the most
common goal of multi scale modeling is to predict the macroscopic
behavior of a process from first principles (up scaling or bottomup
approach). Though multiple scale models are not new, the topic has
recently taken on a new sense of urgency. A number of hybrid
approaches have been created in which ideas coming from distinct
disciplines or modeling approaches have been unified to produce new
and computationally efficient techniques.
This will be a selfcontained course. We will start by reviewing
elements of the homogenization approach, singular perturbation
theory as well as coarsegraining of stochastic processes. Then we
will discuss a number of new ideas that have emerged in the last few
years together with a variety of applications. Students will be
working in groups on several projects and will present them in class
in the end of the course.
Books:
Multiple Scale and Singular Perturbation Methods (Applied Mathematical
Sciences), J.K. Kevorkian and J.D. Cole, Springer (1996).
Special Issue on Multiscale Modeling in Biology, Multiscale Modeling
and Simulation: A SIAM Interdisciplinary Journal, editors: Mark
Alber and Thomas Hou, Volume 3, Number 2 (2005).
Multiscale Methods in Science and Engineering, Lecture Notes in
Computational Science and Engineering, editors: Bjrn Engquist, Per
Ltstedt, and Olof Runborg, Springer (2005).
Selected Final Projects
Click on the picture on the right to download the
final reports.

"An explicit spatial model of yeast
colony growth" by Thomas Apker and Jianfeng Zhu 


Hybrid System: Combing the Ising Model
and an Ordinary Differential Equation by Richard Gejji and
Tanya Kazakova 


Review of "A Multiscale Model for Tumor
Growth" by Matt Rissler Fang Qi 


Multiscale Material Modeling of Fracture
and Crack Propagation by Gilberto MejíaRodríguez and
Chandan K Mozumder 


Experimental and computational
investigations in bone structure and adaptation by Matthew
Landrigan, Charles Penninger, and Marissa J. Post 

