Bjorn Engquist (Princeton)

Heterogeneous multi-scale methods

The heterogeneous multi-scale method (HMM) is a framework for developing and analyzing numerical methods for multi-scale problems. A numerical method on the macro-scale is coupled to micro-scale simulations, which provide data for the macro-scale solver. We will discuss analytic and numerical examples for dynamical systems and homogenization.

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Thomas Y. Hou (Caltech)

Multiscale Computation of Flow Through Heterogeneous Media

Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. Here, we introduce a dynamic multiscale method for computing nonlinear partial differential equations with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions local in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Such approach overcomes the common difficulty associated with the memory effect in deriving the global averaged equations for incompressible flows with multiscale solutions. It provides an effective multiscale numerical method for computing two-phase flow and incompressible Euler and Navier-Stokes equations with multiscale solutions. In a related effort, we introduce a new class of numerical methods to solve the stochastically forced Navier-Stokes equations. We will demonstrate that our numerical method can be used to compute accurately high order statitstical quantites more efficiently than the traditional Monte-Carlo method.

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Frederic Y.M. Wan (UC Irvine)

Morphogen Gradients and Tissue Patterning

Gradients of morphogen concentrations are known to be responsible for cell signaling and tissue patterning. Until the turn of the century, conventional wisdom had been that diffusion would be the main mechanism responsible for the formation of steady gradients. With some experimental results reported in 2000, feasibility of the diffusion mechanism for steady morphogen gradients came under more serious scrutiny and challenge and alternatives such as "transcytosis" and "bucket brigades" began to gain wider acceptance. One type of concern about the role of diffusion emerges from a basic model of reversible binding between receptors and steadily infused morphogens. Even with diffusion, such a model predicts an effectively uniform distribution of moprphogen-receptor complexes (instead of a gradient of such complexes). Another type of concern pertains to the presence of morphogen-receptor complexes in the cell interior (instead of just on the cell surface). In a recent article [1], we presented numerical evidence from more realistic models that should help to alleviate both types of concerns. By allowing for degradation of morphogen-receptor complexes (which is know to take place), numerical results showed useful steady state concentration gradients of receptor - bound morphogens could be formed for suitable combinations of biological parameter values. Also for cases computed, the presence of (biologically realistic) internalization actually helps rather than inhibits the formation of these gradients by diffusion. Time-permitting, this talk will present the followings:

1. The mathematical analysis and computational results that underpin the conclusion in [1]: These include theorems on the existence, uniqueness and stability of steady state gradients when the biologically realistic phenomenon of degradation is included in the model. The necessary and sufficient condition for the existence of a unique steady state is now known to depend only on the ratio of two of the many dimensionless biological parameters.
2. The portability of conclusions based on a one-dimensional model of diffusion-reversible binding-degradation: The conclusions in [1] based on one-dimensional models are representative of the biologically more realistic two- and three-dimensional models except for layer phenomena adjacent to the dorsal-ventril edges.
3. The structural similarity between models with and without internalization: The steady state gradients for the much more complex model that allows for internalization are governed by the same boundary value problem as that for the case without internalization except with the dimensionless parameters replaced by their effective counterparts. While their mathematical structures are not similar, the stability analyses of the two different models have led to results that are also analogous up to the relevant dimensionless biological parameters.
4. Preliminary results for more complex models that include other effects such morphogens binding with other molecules (in addition to receptors) and feedback mechanisms: These results appear to further strengthen the robustness of the role of diffusion on the formation and maintenance of useful steady morphogen gradients.

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Barry Merriman (UC Los Angeles)

A Mathematician's Perspective on Molecular Biology

Coming from the field of applied mathematics, I have spent the past two years working on molecular biology, with the ultimate goal of merging mathematical and molecular techniques to create new tools for investigating biological processes at the "genomic" scale. In this talk I will briefly summarize the present state of genomics (in terms a mathematician can understand), and then attempt to answer the frequently asked question "Where's the math?". Towards this end, I will present my "insider" perspective on the hidden role of mathematics in molecular biology, and illustrate it with examples drawn from the history of molecular biology and from my own current research projects.

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Panel Discussion

The panel discussion will be on applied math education and research, and their impact on the academic or industrial career of our applied math Ph.D. students. Other questions and suggestions are welcome too.

Tom Chou (UCLA)

A Length-dynamic Tonks gas model of histone isotherms:

A model for the adsorption of interacting histone proteins on DNA substrate is presented. The statistical mechanics of the system is mapped onto a one-dimensional Tonks gas model, and solved exactly. DNA wrapping, coverage, protein densities, and their fluctuations are computed.

Harish Bhat and Jimmy Fung, Caltech

Charged Particle Motion: Averaging Techniques

The motion of charged particles in a magnetic field is examined as a model problem for averaging techniques. A coarse trajectory of this system is calculated by coupling a computational superstructure to a fine-scale numerical integrator. Models for the average motion may also be derived from an averaged action principle. Existing approaches, such as the guiding center approximation, are reviewed, and a new model is proposed.

Alfonso Limon (Claremont Graduate University)

A New Algorithm for Capacitance Tomography Imaging of Two-phase Flow Regimes

The Instituto Mexicano del Petroleo has been doing extensive research in capacitance tomography as a viable method for determining volume fractions and velocity measurements for multi-component flows in oil pipelines since 2000. In an effort to expand the computational tools utilized to solve this problem, the Math Clinic at Claremont Graduate University studied this project during the 2002-2003 academic year. In this poster the physical set-up of the measurement section of the pipeline will be described together with the current algorithms for extraction of the oil flow variables. An introduction to a curvilinear-coordinate-based algorithm for the forward problem will be presented, in addition to an adaptive direct/iterative approach to solve the forward problem efficiently. The inverse problem will be discussed in the framework of two regularization methods: the total variation regularization and the semi-H1 norm regularization. The major difference between the two r! egularizations methods is that the total variation does not smooth jump-discontinuities. Performance comparisons between our code and the finite-element based package EIT2D will be presented.

Cem Mutlugun (Claremont Graduate University)

The standard transistor commonly used in the semi-conductor industry for developing integrated circuits is known as a MOSFET (metal-oxide-silicon-field-effect transistor). Integrated circuit (IC) properties are simulated with a program whose generic name is SPICE (Simulation Program with Integrated Circuit Emphasis). SPICE is a simulator used to model (millions of) electrical circuits at the transistor level. Because of the numerous transistors typically associated with ICs, SPICE requires simple mathematical expressions for the current/voltage relations characterizing each transistor. The region of the transistor that we are investigating consists of the gate, insulator, and base. The interaction between these regions in terms of the electical potential variations in each, due to an applied gate voltage, is expressed in terms of a capacitance in SPICE. The objective of this study is to improve the approximations for modeling gate capacitance in metal oxide! semiconductor (MOS) and poly-silicon (POS) devices for better comparisons with empirical data and for implementation into SPICE simulations.

Lyle Noakes (Department of Mathematics and Statistics, The University of Western Australia)

"Numerical Methods for Interpolating Rotations"

The problem of interpolating rotations of Euclidean 3-space $E^3$ is much more complicated than interpolation in $E^3$ itself. We review some of the available approaches, including recent advances in computational methods.

Surya Harith Vanaparthy (Department of Mechanical and Environmental engineering, University of California, Santa Barbara)

"Density-driven instabilities of miscible fluids in a capillary tube: Linear Stability analysis"

A linear stability analysis is presented for the miscible interface formed by placing a heavier fluid above a lighter one in a vertically oriented capillary tube. The analysis is based on the three-dimensional Stokes equations, coupled to a convection-diffusion equation for the concentration field, in cylindrical coordinates. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate as well as a Rayleigh number and a dimensionless interfacial thickness.

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