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Applied Math. Seminar, Spring 2008

**January 28**

Speaker: Rob MacLeod, University of Utah, Scientific Computing and Imaging Institute (SCI) and

Cardiovascular Research and Training Institute (CVRTI)

**Title: ** Simulation of Defibrillation: A Little Math Goes a Long Way

**Abstract: **
Although implantable cardiac defibrillators (ICDs) have been available since
1980, the placement of the devices in patients has remained largely an
empirical art based on animal studies and clinical experience. The resulting
standard placements work effectively but are often unsuitable for use in
children, because of factors like the small size of the torso, the subsequent
growth of the child, and the unusual anatomies of most pediatric ICD
recipients. Children receiving ICDs typically have congenital heart defects
leading to surgical correction and the electrical instabilities that require
the use of such a device.

This clinical need motivated a study, the goal of which was to create a
simulation system that allows physicians to evaluate ICD placement in pediatric
patients using a patient specific mathematical modeling approach. The
simulations are based on subject specific geometric models derived from CT and
MRI scans of the thorax, in which we embed the ICD device and associated
electrodes. A user can adjust electrode locations interactively and then
evaluate the effectiveness of each configuration. The simulation is a finite
element method solution to a Poisson's equation for electrostatic potential.
The study has so far generated very encouraging results, even under simplifying
assumptions, which provides additional motivation to begin testing the system
in clinical cases.

~~February 1, 3:15pm-4:15pm: SPECIAL DATE, TIME AND LOCATION (LCB 222)~~

**DELAYED until further notice.**

Speaker: Dmitri Vainchtein, Georgia Institute of Technology, Center for Nonlinear Science

**Title: **Resonances-induced chaotic advection in a cellular flow

**Abstract: **In my talk I present a quantitative theory of resonance-induced chaotic advection and mixing in time-dependent volume-preserving 3D flows using a model cellular flow introduced in [T. Solomon and I. Mezic, Nature, 425, 376 (2003)] as an example. Specifically I show that chaotic advection is dramatically enhanced by a time-dependent perturbation for certain resonant frequencies. I compute the fraction of the total volume of the cell that participates in mixing as a function of the frequency of the perturbation and show that at resonance essentially complete mixing in 3D can be achieved.

~~February 11~~ DATE CHANGE TBA

Speaker: Peg Howland, Utah State University, Department of Mathematics and Statistics

**Title: ** TBA

**Abstract: ** TBA

__ February 20__ (

Speaker: Dong Li, Institute for Advanced Studies

**February 25**

Speaker: Samuel Isaacson, University of Utah, Mathematics Department

**Title: ** Connections between the Reaction-Diffusion Master Equation, Quantum Field Theory, and Scattering

**Abstract: ** We will explain how the reaction-diffusion master equation (RDME) may be mapped to a lattice quantum field theory. The approach we take will parallel that developed by Doi (J. Phys. A: Math Gen. 1976) for classical many particle systems, and complement the mapping of the RDME developed by Peliti (J. Physique 1985). We will also discuss how the formal continuum limit of the RDME, when rigorously defined, may be interpreted as a coupled system of diffusion equations with pseudo-potential interactions. Pseudo-potentials were first used by Fermi as a method for approximating hard-core scattering problems in quantum mechanics. We will show how the pseudo-potential model gives an asymptotic approximation to a model of Smoluchowski.

**March 3**

Speaker: Rebecca Brannon, University of Utah, Department of Mechanical Engineering

**Title: **Mathematical challenges in modeling high-rate failure

**Abstract: **Under conditions of unstable dynamic fragmentation,
microscale variability can not be "smeared" at the continuum scale as
it can under stable loading. Microscale heterogeneity not only causes a
homogeneously loaded to sample to fail inhomogeneously, but it also causes
small samples to be stronger, on average, than large samples. Accounting for
the effects of micro-heterogeneity by imposing uncertainty and scale effects
within an otherwise conventional engineering damage model will be shown to
mitigate mesh-sensitivity and dramatically improve results in dynamic
indentation simulations. For simulating failure of diametrically loaded disks,
this statistical scale-dependent theory matches observed trends in strength
data, but (in contrast to the indentation simulations) mesh sensitivity is
actually exacerbated, which not only disallows quantitative parameter fitting
but also shows that success in one problem does not ensure improvements in
other problems. Noting that coarsening induced by low-order basis functions
might be the source the mesh sensitivity, modified shape functions (similar to
up-winding) have been developed for one-dimensional problems. However,
generalization to higher dimensions is unclear. For problems involving massive
material deformations, so-called artificial healing and other advection-induced
corruptions of the material state fuel research in particle methods as an
alternative to traditional computational methods. Particle methods eliminate
advection errors, but at the expense of accuracy in the momentum solver. Basic
mathematical theory for Utah's MPM particle code will be discussed in the
context of the need for efficient and accurate mathematical approaches to
minimizing momentum errors when particles cross cell boundaries. As time
allows, various other mathematical challenges in high-rate large-deformation
mechanics will be discussed.

(PDF version)

**March 24**

Speaker: Eddie Wadbro, Uppsala University, Uppsala, Sweden

**Title: ** Design optimization for wave propagation problems

**Abstract: ** Using gradient-based optimization combined with numerical
solutions of the Helmholtz equation, we successfully design an
acoustic device with high transmission efficiency and even
directivity throughout a two-octave-wide frequency range. The
device consists of a horn, whose flare shape is subject to
optimization, together with an area in front of the horn where
solid material arbitrarily can be distributed using topology
optimization techniques, effectively creating an acoustic lens.
Similar techniques can also be used to attack the inverse problem
assiciated with microwave tomography. That is, reconstructing the
dielectric properties of lossy objects using microwave radiation
and measurements of the scattered field. Important physiological
conditions of living tissues, such as blood flow reduction and
the presence of malignant tissue, are accompanied by changes in
their dielectric properties. In the final part of my talk I
describe how the computational power of a modern graphics card
can be used to speed up the computations for a typical pixel
based material distribution problem, enabling the solution of a
constrained nonlinear optimization problem with over 4 million
descision variables.

**March 31**

Speaker: Ilya Krishtal, Northern Illinois University, Department of Mathematical Sciences

**Title: ** Wiener's Lemma in Frame Analysis

**Abstract: ** In the first part of the talk I will show how abstract
Banach algebra and harmonic analysis techniques lead to a sweeping
generalization of the famous Wiener's *1/f* Lemma. In the second part,
the above generalization will be used to explain localization results
for dual Gabor frames. In the end of the talk, if time permits, other
applications will be discussed. These may include results on spectral
theory of time-frequency shifts and regular factorizations of
pseudo-differential operators. Presented results were obtained in
collaboration with R. Balan and K. Okoudjou.

**April 7**

Speaker: Yuliya Gorb, Texas A&M University, Department of Mathematics

**Title: ** Singular Behavior of the Overall Viscous Dissipation Rate of
Highly Concentrated Suspensions

**Abstract: ** We present a two-dimensional mathematical model of a highly
concentrated suspension of rigid particles close to touching in an
incompressible Newtonian fluid. The overall viscous dissipation rate of such a
suspension exhibits a singular behavior. The objectives of our study are
two-fold: (i) to obtain all singular terms in the asymptotics of the overall
viscous dissipation rate as an interparticle distance parameter tends to zero,
(ii) to obtain a qualitative description of a microflow between
neighboring particles in the suspension. Our analysis provides the limits
of validity of two-dimensional models for three-dimensional problems and
highlights novel features of two-dimensional physical problems (e.g. thin
films). It reveals that the Poiseuille type microflow contributes to
a singularity of the dissipation rate. We show that that under certain
conditions the model exhibits an anomalously strong rate of blow up when
the concentration of particles tends to maximal.

**April 14**

Speaker: Joe Koebbe, Utah State University, Department of Mathematics and Statistics

**Title: ** Construction of Adaptive Wavelets Using Differential Operators

**Abstract: ** The talk will show how to construct adaptive wavelets based
on properties of partial differential operators in homogenization applications
and approximate solution of conservation laws. Both constructions require the
development of a nonlinear transform. These will be presented in detail.

**April 21**

Speaker: Kenneth Kuttler, Brigham Young University, Mathematics Department

**Title: ** Problems involving damage

**Abstract: ** I will give a summary of a few problems which involve a damage
parameter as well as a short description of the physical motivation. Then I
will mention methods which have successfully resolved the mathematical
theory in some cases. In conclusion, I mention some unsolved problems.

__ May 8 (Thursday)__, LCB 215, 4:15pm

Speaker: Pierre Seppecher, Université de Toulon et du Var, Institut de Mathématiques de Toulon

__ May 12__, LCB 215, 4:15pm

Speaker: Viet Ha Hoang, University of Cambridge, Dept. of Applied Mathematics and Theoretical physics

155 South 1400 East, Room 233, Salt Lake City, UT
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