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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.

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

February 20 (Wednesday: 4:15pm-5:15pm, Location JWB 335)
Speaker: Dong Li, Institute for Advanced Studies
Title: The characterization of minimal mass blow up solution of focusing mass-critical nonlinear Schrödinger equations
Abstract: Let u be a global solution to the focusing mass-critical nonlinear Schrödinger equation for radial symmetric H1 initial data with ground state mass in dimension d ≥ 4. We prove that if u does not scatter, then up to phase rotation and scaling, u is the solitary wave eitQ where Q is the ground state. This together with the results from F. Merle in \cite{merle} shows that the pseudo-conformal blow up and the solitary wave are the only two minimal mass blow up solutions.

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
Title: 3D-2D analysis for the optimal elastic compliance problem
Abstract: A prescribed amount of linear elastic material has to be placed in a design region of very small height in order to maximize the resistance of the plate. We prove that, for the optimal shape and at the limit when the height tends to zero, flexion and extension are coupled through a Kirchhoff-Love motion. We give optimality conditions and find that the (rescaled) optimal shape has a disconnected section. The results differ fundamentally from the results obtained by optimizing the thickness of a plate under the constraint of a connected section.

May 12, LCB 215, 4:15pm
Speaker: Viet Ha Hoang, University of Cambridge, Dept. of Applied Mathematics and Theoretical physics
Title: Sparse Finite Element Method for Nonlinear Elliptic Problems with Multiple Scales
Abstract: A sparse tensor product Finite Element (FE) method is developed for the high-dimensional limiting problem obtained by applying the multiscale convergence to a multiscale elliptic problem in Rd. The limiting problem is posed in a product space, so tensor product FE spaces are used for discretization. This sparse FE method requires essentially the same number of degrees of freedom to achieve essentially equal accuracy to that of a standard FE scheme for a partial differential equation in Rd. Multiple scale linear elliptic problems and nonlinear monotone problems are considered. In many cases, it is shown that the solution of the high-dimensional limiting problem is smooth. This leads to an analytic homogenization error, which together with the FE error provides an explicit error estimate for an approximation to the solution of the original multiscale problem. Without this regularity, such an approximation always exists when the meshsize and the micro scale converge to 0, but without a rate of convergence.

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