Department of Mathematics

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Past talks for Spring 2012

January 23
Speaker: Frank Stenger, University of Utah, School of Computing
Title: Solving the Helmholtz Equation for Ultrasonic Tomography

February 6
Speaker: Justin Kao, Dept. of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology
Title: Inclusions in fluid coating and solidification
Abstract: Fluid coatings occur in a wide variety of situations, from the manufacture of semiconductors to the foam lacing left behind after drinking a pint of beer. Sometimes coating uniformity is desirable, but in many cases, these coatings are heterogeneous, with either defects or deliberate inclusions. In this talk I examine the deposition of isolated bubbles in Landau-Levich (dip coating) flow, and the self-assembly of particles in Landau-Levich flow of a suspension. These phenomena are explained through a combination of modeling, experiment, and analysis.
Another fluid flow involving inclusions is the capture of foreign particles during solidification of liquid melts, a problem relevant to situations such manufacturing of composite materials and cryopreservation of cells. Particle behavior in this system is governed by a critical, threshold solidification rate. I describe modeling and numerical computations of particle capture in pure and binary melts.

February 10 (FRIDAY 4-5pm -- LCB 225 -- joint with Mechanical Engineering)
Speaker: Ole Sigmund, Technical University of Denmark, Dept. of Mechanical Engineering
Title: Topology Optimization with applications in material and wave-propagation problems.
Abstract: The topology optimization method, originally developed for weight-optimal design of automotive and aerospace structures by Bendsøe and Kikuchi (1988), has in recent years become a popular tool for the systematic design of structures and materials in a wide range of physical disciplines. The presentation will give an overview of recent developments within topology optimization and its applications, with special emphasis on extremal material design and wave propagation problems in acoustics and optics.
Topology optimization provides optimal material distributions for various problems that can be modeled using finite element or finite difference analysis. Element or node-wise material densities constitute the design variables and the deterministic optimization procedure consists of repeated (finite element/difference) analysis steps, analytical gradient evaluations and mathematical programming-based design updates. Typically, the method requires a few hundred function evaluations (i.e. finite element analyses) to converge. Recent developments include consideration of manufacturing uncertainties in the optimization process.
The presentation will include studies on the design of extremal materials with negative Poisson's ratios and maximal damping. Furthermore, the presentation will show studies on the manipulation of waves by acoustic and optical cloaks as well as by structured surfaces that change color appearance.

February 13
Speaker: Benjamin Webb, Brigham Young University, Mathematics dept.
Title: Consolidating Information in Dynamical Networks
Abstract: A major obstacle in understanding the dynamic behaviour of a network is that the information needed to do so is spread throughout the various network components. Because of this it is tempting to find ways of concentrating this information while preserving some fundamental property or features of the system's dynamics. With this in mind we introduce the concept of an isospectral network expansion. The idea behind such expansions is that a network's structure can be modified in a number of ways that preserve the system's dynamics while simultaneously concentrating the network's structural/dynamic information. This method allows us to give improved estimates for determining whether a network exhibits relatively simple dynamics.

February 15 (WEDNESDAY 3:05pm - LCB 323 - Math Bio seminar)
Speaker: Dharshi Devendran, Courant Institute for Mathematical Sciences, NYU
Title: Applications and Theory of a Continuum-Mechanics-Based Immersed Boundary Method
Abstract: Fluid-structure interaction problems, in which the dynamics of a deformable structure is coupled to the dynamics of a fluid, are prevalent in biology. For example, the heart can be modeled as an elastic boundary that interacts with the blood circulating through it. The immersed boundary method is a popular method for simulating fluid-structure problems. The traditional immersed boundary method discretizes the elastic structure using a network of springs. This makes it difficult to use material models from continuum mechanics within the framework of the immersed boundary method. In this talk, I present a new immersed boundary method that uses continuum mechanics to discretize the elastic structure, with a finite-element-like discretization. This method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. Next, we apply the method to a three-dimensional fluid-structure interaction problem. Finally, I will present theory for this new immersed boundary method.

February 17 (FRIDAY, LCB 222, 4-5pm)
Speaker: Jianfeng Lu, Courant Institute of Mathematical Sciences, New York University
Title: Metastability and coarse-graining of stochastic systems
Abstract: The study of rare events in physical, chemical and biological systems are important and challenging due to the huge span of time scales. Coarse-graining techniques, Markov state models for example, are employed to reduce the degree of freedom of the system, and hence enables simulation and understanding of the system on a long time scale. In this talk, we will introduce a novel construction of Markov state model based on milestoning. We will focus on the analysis of quality of approximation when the original system is metastable. The analysis identifies quantitative criteria which enable automatic identification of metastable sets.

February 27
Speaker: Sarang Joshi, SCI, University of Utah
Title: Computational Anatomy: Simple Statistics on Interesting Spaces for Developing Imaging Biomarkers Analysis
Abstract: A primary goal of Computational Anatomy is the statistical analysis of anatomical variability. Large Deformation Diffeomorphic transformations have been shown to accommodate the geometric variability but performing statistics of Diffeomorphic transformations remains a challenge. I will start with the simple concept of defining the "Average Anatomy" and then extend this to the study of regression and co-variation of anatomical shape with independent variables. The motivation is to model the inherent relation between anatomical shape and clinical measures and evaluate its statistical significance. We use Partial Least Squares for the multivariate statistical analysis of the deformation momenta under the Large Deformation Diffeomorphic framework. The statistical methodology extracts pertinent directions in the momenta space and the clinical response space in terms of latent variables. We report the results of this analysis on 313 subjects from the Mild Cognitive Impairment group in the Alzheimer's Disease Neuroimaging Initiative (ADNI).

March 5
Speaker: Arnold D. Kim, University of California at Merced
Title: Backscattering of polarized beams by layered tissues
Abstract: A layered tissue model has a thin, epithelial layer supported underneath by a thick, stromal layer. Each layer has its own optical properties which, in turn, provide useful medical diagnostic information. We present an asymptotic analysis of the boundary value problem for the vector radiative transport equation that governs a polarized beam incident on layered tissues. In doing so, we are able to propose novel methods to study the optical properties of epithelial tissue layers which have applications in the early diagnosis of cancer.
This work involves collaborations with Miguel Moscoso, Shelley Rohde, and Julia Clark.

March 7 (Special day and time: Wed 2pm-3pm -- LCB 323)
Speaker: Yi-Ping Ma, University of Chicago, Dept. of Geophysical Sciences
Title: Applications of pattern formation: supraglacial lakes and spatially localized states
Abstract: I will start with an overview of methods in the mathematical theory of pattern formation, and proceed to introduce two examples where these ideas can be applied. As the first example, I will describe our recent attempts to explain supraglacial lake patterns on the George VI ice shelf in Antarctica, based on an asymptotic theory of viscous buckling. As the second example, I will provide an overview of recent development in the study of spatially localized states in one and two spatial dimensions. These localized states are observed in physical systems ranging from ferrofluids to vibrated granular media, and often result from bistability between a featureless state and a patterned state. I will emphasize insights gained from the study of two nonlinear PDEs, namely the quadratic-cubic Swift-Hohenberg equation and the harmonically (or 1:1) forced complex Ginzburg-Landau equation. It is shown that the nature of the bistability determines the spatial bifurcation structures and temporal dynamics of spatially localized states.

March 19
Speaker: Alexander V. Mamonov, University of Texas at Austin, ICES
Title: Point source identification in non-linear advection-diffusion-reaction systems
Abstract: We consider a problem of identification of point sources in time dependent advection-diffusion systems with a non-linear reaction term. The linear counterpart of the problem in question can be reduced to solving a system of non-linear algebraic equations via the use of adjoint equations. We extend this approach by constructing an algorithm that solves the problem iteratively to account for the non-linearity of the reaction term. We study the question of improving the quality of source identification by adding more measurements adaptively using the solution obtained previously with a smaller number of measurements.

March 26
Speaker: Yaniv Gur, SCI, University of Utah
Title: Higher-order tensor decompositions in neuroimaging
Abstract: Higher-order tensors are considered as multidimensional arrays or N-mode arrays with N>2. There are various decompositions associated with higher-order tensors which are widely used now to solve problems in different fields such as genomic signal processing, computer vision, data mining, and much more. The use of these decompositions is constantly expanding to new applications. In this talk I will focus on low-rank tensor approximations via the CANDECOMP/PARAFAC (CP) decomposition which decomposes a tensor as a sum of rank-one tensors and can be considered to be higher-order extension of the matrix singular value decomposition. I will give an introduction to diffusion MRI and show how higher-order tensors and CP decompositions can be used to resolve white-matter structure of the brain.
A joint work with Chris Johnson, Sarang Joshi and Fangxiang Jiao.

April 4 (Wed 3:05pm LCB 323 - joint with Math BIO)
Speaker: Qinghai Zhang, University of California at Davis
Title: Resolving the boundary layer on your laptop: a fourth-order projection method for adaptively solving the Navier-Stokes equations.
Abstract: A projection method with fourth-order accuracy both in time and space will be presented for solving the incompressible Navier-Stokes equations on periodic and no-slip domains with adaptive mesh refinement and parallel computing. Spatial discretization employs classical finite volume stencils while temporal integration adopts a semi-implicit, L-stable additive Runge-Kutta (ARK) method which treats the non-stiff convection term explicitly and the stiff diffusion term implicitly. The resulting Poisson- and Helmholtz-type linear systems are solved with an efficient multigrid algorithm. The central difficulty on no-slip domains, i.e. the non-commutativity of the projection and Laplacian, is handled by solving for the Stokes pressure and add its gradient back to complete the evolution of the velocity. The well-posedness and stability of this approach is confirmed by a recent analysis on the bound of the Laplace-Leray commutator. Results from a number of benchmark test problems show that the ratio of the CPU time and memory expense of the fourth-order method to those of previous second-order methods is less than 1/500. The efficiency is further enhanced by adaptive mesh refinement for additional power of resolving fine structures. I will also discuss the generalization of this approach to complex multi-phase flows such as free-surface flows and fluid-structure interaction.

April 23
Speaker: David Fullwood, Brigham Young University, Mechanical Engineering
Title: Mathematical Tools in the Description and Design of Microstructure
Abstract: The description, analysis and design of microstructure require efficient metrics that reflect the structural attributes that underpin the material properties of interest. These attributes vary from connectivity to periodicity and clustering, thus requiring a suite of mathematical tools that capture geometry in different ways. Computational homology is one tool that has recently shown promise for encapsulating the topological and connected nature that influences stiffness and conductivity, for example. On the other hand, n-point correlation functions are notoriously poor at capturing connectivity, but fit nicely with a range of structure-property relations. This talk will discuss current mathematical tools employed in the description of microstructure, and highlight some of the limitations that cry-out for new developments in this area.

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