Applied Mathematics Colloquia Series
Archive

Fall 2004

September

09/03
Jay R. Walton
Department of Mathematics and Aerospace Engineering
Texas A&M University

Mathematical Modeling of Cardio-Vascular Disease

Cardiovascular disease remains the leading cause of death in developed countries. Recently significant progress has been achieved in understanding the role of chronic inflammation of the endothelial or inner most layer of the arterial wall in the onset and progression of the disease and the many biochemical pathways leading to the genesis of atherosclerotic lesions. In this talk, I present recent modeling attempts of the main processes behind atherogenesis. A surprisingly simple system of six reaction/diffusion equations of the Keller-Segal chemotaxic type for three cellular species and three chemical species seems to capture many of the important features observed in atherogenesis. On a fixed domain, this system can exhibit rapid blow-up in finite time after a threshold has been surpassed which can be interpreted as a runaway diseased state. This blow-up is of the Dirac point mass type. The free boundary value problem for this system corresponding to the growth and encroachment of the lesion into the arterial lumen (interior flow region) is also discussed.

A second problem area to be discussed concerns modeling growth and remodeling of the medial (central muscular) layer of an arterial wall due to chronic hypertension. The model differentiates between three type of growth processes, hypertrophy in which smooth muscles grow is mass through either the production of intracellular proteins or extracellular matrix proteins and hyperplasia or smooth muscle proliferation. A key step in the modeling involves a pair of partial differential equations, one non-linear parabolic and the other non-linear hyperbolic, keeping track of both the number density of smooth muscle cells and their average mass. The issue of traveling wave solutions for this system is discussed.

09/10
Paul Fischer
Mathematics & Computer Science Division
Argonne National Lab

Current Trends in High-Order Numerical Methods for Scientific Simulation

As scientific simulation continues to expand, high-order discretizations are finding use in increasingly diverse applications featuring broad ranges of multiphysics and multiscale phenomena. In addition to their attractive numerical properties, high-order weighted residual techniques (WRTs), which yield significant data re-use and minimal interprocessor communication, are readily able to exploit high-performance CPUs on modern supercomputer architectures that comprise cache-based microprocessors interconnected by a relatively slow network. In the first half of this talk, we present a brief overview of current trends for high-order methods in scientific simulation. We touch on different approaches to WRTs, including the spectral element method, the discontinuous Galerkin method, and the p-type finite element method, as applied to problems in electromagnetics, fusion, geophysics, and fluid mechanics, and we discuss several algorithmic features common to these methods.

In the second half of this talk, we consider applications of the spectral element method to simulation of high Reynolds number incompressible flows in complex domains. The spectral element method is based on rapidly convergent, high-order WRTs employing tensor-product polynomial bases of degree N in each of E deformed quadrilateral or hexahedral elements. Because of its low numerical dissipation and dispersion, the method is well suited to simulation of flows at transitional Reynolds numbers, where physical viscosity is small and integration times are relatively long. We present an overview of the time-stepping scheme and iterative solvers, with some emphasis on the use of projection methods to accelerate convergence and reduce communication overhead. We close with several example applications, including the simulation of transition in vascular flows.

09/24
John Sutter
United States Geological Survey

The Earth Surface Processes Research Institute (ESPRI): A Joint Venture between the UA and the U.S. Geological Survey

Understanding the response of earth-surface processes to both natural and potentially human-induced environmental changes constitutes the paramount scientific challenge to earth sciences in the 21st century. In response to this challenge, the U.S. Geological Survey (USGS) and the University of Arizona (UA) are jointly establishing the Earth Surface Processes Research Institute (ESPRI) on the University of Arizona campus in Tucson. The purpose of this institute is to provide a venue for groundbreaking research on landscape change and ecosystems response, with a specific emphasis on arid and semi-arid environments. Researchers from multiple disciplines within one institute, forging the widest possible perspective, can address uniquely questions of applied and fundamental value. ESPRI is being developed around four core research areas:

(1) Quaternary bio/geochronology - to develop accurate and precise means to determine the age of an event at various scales over the past million years.

(2) Quantitative geomorphology - by using accurate measures of time, determine the rates of processes that modify the surface of the solid earth.

(3) Quantitative hydrogeology - model processes and process rates that affect water availability at the surface, and, therefore to the geomorphic processes.

(4) Forecasting landscape change and ecosystem response - using accurate process rates, forecast effects of climate variability and of anthropogenic events on the landscape and ecosystems.

Short vignettes from projects will be used to characterize each of these areas of research.

October

10/08
Todd Squires
Department of Applied & Computational Mathematics
California Institute of Technology

Using the inner ear to balance: Why it works so well and what can go wrong

In order to sense rotation, all vertebrates use a set of inner-ear organscalled semicircular canals. These exquisite little structures are almost entirely mechanical in origin, and rely on physical displacements of fluids and an elastic membrane to translate a rotation into a neural signal. I briefly review the basic operation of the semicircular canals, which can be easily understood in terms of simplified physical processes. I will then discuss research into two areas:

1) "Top shelf vertigo", or BPPV, is a mechanical disorder of the semicircular canals. Persons with BPPV experience severe dizziness after their head is tilted back (as if to look at the top shelf). The medical community has proposed that rogue crystals in the canal are to blame. We quantitatively examine such proposals from a fluid mechanical standpoint. (with H. Stone, T. Hain and M. Weidman)

2) Shocking fact: the semicircular canals of all vertebrates -- from mice to whales -- are essentially the same size. Why would this be? If one thinks of evolution as an ongoing search to improve, then perhaps the semicircular canals are `optimized'. After all, the eye operates at the single-photon level and the cochlea is thermal-noise limited. In fact, the "optimal design" for the semicircular canals has dimensions that are consistent with those shared by all vertebrates, and parameter space is quite constrained.

10/15
John McLean
Arete Associates

Estimation of bathymetry and current from ocean wave spectra using passive optical sensors

Natural light from the ocean is modulated by waves, and one can infer ocean properties from optical images of the sea surface. By dwelling on a particular patch of the ocean, the full 3-D surface wave spectra (2 spatial dimensions + time) may be computed. Using well-known properties of wave propagation (i.e. the dispersion relation), one can accurately estimate both bathymetry and current from the wave spectra. This presentation will describe the theoretical basis of the remote sensing technique, describe the measurement system, and present results from recent field tests.

Bio: John McLean is president and CEO of Arete Associates, a small (~200 person) company that specializes in ocean remote sensing. Dr. McLean’s research interests include wave propagation and stability, optical remote sensing, and high resolution imaging systems. Dr. McLean graduated from the University of Arizona (BS Engineering Math) in 1975, and received a PhD in Applied Math from Caltech in 1980.

10/22
Doron Zeilberger
Department of Mathematics
Rutgers University

Hypergeometrics Across the Mathematical Curriculum

Hypergeometric functions and series are almost everywhere dense in mathematics, from the purest Pure Math to the most applied Applied Math. This will be surveyed, culminating with very recent exciting work by Robert Maier (UA), who found many new hypergeometric series transformations that Edouard Goursat, more than 120 years ago, completely missed.silico tumor simulator. Our goal is to customize cancer drug therapy in the

10/29
Vittorio Cristini
Department of Mathematics
UC Irvine

Biocomputational and Experimental Modeling of Cancer and Chemotherapy

This research focuses on the biocomputational and experimental modeling of cancer tumor growth and therapy. To this end we have created a sophisticated in silico tumor simulator. Our goal is to customize cancer drug therapy in the clinical setting by using tumor information specific to each patient. This approach should not only save time and resources in cancer treatment, but also be most beneficial to the patient.

Our multiscale, multidimensional tumor simulator has the capability of showing cancer progression through the stages of diffusion-limited dormancy, vascularization and rapid growth, new equilibrium, and tissue invasion. This simulator encompasses some of the main physical laws of cancer growth and creates an in silico system that exhibits combined two-dimensional tumor growth and angiogenesis. The system captures the complicated morphology and connectedness at the tumor/tissue interface, including invasive fingering, tumor fragmentation, and healthy tissue degradation. Angiogenesis is included as a continuous feedback process involving tissue growth and nutrient demand.

Here, we use the tumor simulator to demonstrate fundamental transport limitations in delivering anticancer drug into tumors, whether this delivery is via free drug administration or via nanoparticles injected into the bloodstream. In the case of nanoparticle delivery, making some assumptions regarding targeted delivery, we find that host tissue toxicity can be less than that of traditional delivery, and also further investigate the effect of anti-angiogenic "normalizing" in ameliorating transport limitations. Our results indicate that fundamental transport limitations apply to both traditional and nanoparticle drug delivery. Even in a best-case scenario involving an homogenous tumor of one drug-sensitive cell type, targeted nanoparticle delivery, low host tissue toxicity, and sufficient drug concentration to rapidly kill all cells in vitro, the in vivo rate of tumor shrinkage can be as low as several orders of magnitude less, and the tumor may achieve a new mass equilibrium far above detectable levels.

In collaboration with the UCIMC, we are developing experimental models to study the effects of chemotherapy on specific in vitro and in vivo cancer tumor tissues. This work will yield data that will be used as input to the tumor simulator and provide a sound biological foundation for thebiocomputational model.

November

11/05
Lai-Sang Young
Courant Institute
New York University

Nonequilibrium Dynamics

This talk is about certain simple dynamical models that are out of equilibrium. Our models are made up of chains of identical units - which can be stochastic or Hamiltonian systems - coupled to heat baths at different temperatures at the two ends. Of primary interest are energy profiles and tracer densities at steady state. I will discuss some simple ideas that under suitable conditions allow one to predict these global profiles from the microscopic rules that define the system.

11/19
Olgica Milenkovic
Engineering Center
University of Colorado

Mathematical Problems in DNA Sequence Analysis and Applications

Genomic data analysis is currently one of the fastest growing scientific fields and the focal research problem of a large expert group in molecular biology, mathematics, computer science and coding theory. Genomic data analysis does not only provide partial answers to complex biological problems such as the evolution pathway of our specie or genetic disease treatment, but it also introduces some new research topics in applied mathematics, information and coding theory, as well as electrical and biological engineering in general.

The goal of this talk is to introduce several mathematical problems arising from the area of molecular biology, DNA compression and DNA computing, and to show how these can be approached by using well developed techniques borrowed from statistics, combinatorics, information and coding theory. In this context, we will discuss some new ideas and results regarding:

  1. Random Boolean Function Networks (RBFN) for modeling patterns of gene interactions and their connection to codes on graphs; the treatment of this subject involves some concepts from dynamical systems theory and error-control coding theory.
  2. Statistical DNA analysis techniques and DNA distance measures with application to wavelet or grammar-based DNA compression; the treatment of this subject is based on ideas from classical source coding and fractal sequence analysis.
  3. Coding for DNA computing, including the design of DNA codes with constant GC-content and the reverse-complement property, as well as codes for DNA microarrays; the treatment of this subject is based on classical results from combinatorics and algebraic coding theory.

This is a joint work with B. Vasic, University of Arizona, Tucson.

December

12/03
Roman Polyak
Systems Engineering & Operations Research
George Mason University

Recent Advances in Constrained Optimization

The last 15 years were marked with substantial progress in constrained optimization due to the well known developments in the Interior Point Methods (IPM). We will discuss an alternative to the IPM approach to constrained optimization based on the Nonlinear Rescaling principle. The NR approach consists of rescaling the objective function and/or the constraints of a given constrained optimization problem into an equivalent one and using the Lagrangian for the equivalent problem for both theoretical analysis and numerical methods. The NR method alternates unconstrained minimization of the Lagrangian for the equivalent problem in primal space with both the Lagrange multipliers and scaling parameters update. Our main concentration is the Primal-Dual NR (PDNR) method. It replaces the NR step by solving a particular Primal-Dual (PD) system of equations. Application of the Newton method to PD system leads to the PDNR method that for convex optimization converges under minimum assumptions on the input data. We will show that under the standard second order optimality condition it converges with QUADRATIC rate. Application of the PDNR for solving large scale real life constrained optimization problems and correspondent numerical results will be discussed.