Math 80770, Fall 2006

Multi scale modeling

MWF 1:55 pm - 2:45 pm, Edward J. DeBartolo Hall 240

Instructor: Mark Alber (631-8371),

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 micro-scales to macro-scales 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 bottom-up 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 self-contained course. We will start by reviewing elements of the homogenization approach, singular perturbation theory as well as coarse-graining 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

Multi-scale Material Modeling of Fracture and Crack Propagation by Gilberto Mejía-Rodríguez and Chandan K Mozumder

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