Department of Mathematics

Applied Math. Seminar, Spring 2007

January 8:
Speaker: Neal Patwari, U. of Utah - Department of Electrical and Computer Engineering
Title: Sensing and Processing Across Networks
Abstract: Networking hundreds or thousands or more sensors together in order to accomplish some estimation or monitoring task can be a complicated problem. When these sensors have low computational complexity, energy, and communication bandwidth, this problem becomes much more challenging. Furthermore, we may not have a good model for the behavior of the parameters we wish to estimate, so model-based approaches may not be the appropriate method. Finally, data from each sensor may be very high-dimensional, in particular when sensors are measuring signals or images over periods of time. In this talk, we will discuss particularly useful tools for non-linear dimensionality reduction, or `manifold learning'. We'll review some existing methods and introduce our own. Finally, we'll apply manifold learning to some practical problems:
(1) Estimating the location of wireless sensors in a network,
(2) Anomaly detection of statistically unusual packet traffic on Internet backbone networks,
(3) Multi-static RF motion detection for in-building radar and imaging.

January 17: SPECIAL DATE
Speaker: Alexander Roytburd, U. of Maryland at College Park - Dept. of Materials Science and Engineering
Title: Formation and Magneto-Electric Coupling of Self-Assembled Multiferroic Thin Film Nanostructures.
Abstract: Experimental examples and theories of the formation and electro-magnetic coupling of self-assembled nanostructures consisting of ferromagnetic and ferroelectric phases are presented. The nanostructure architectures are determined by minimizing the free energy which includes the elastic energy of epitaxial stresses arising during the growth of a nanostructured two phase film. Magnetic response to the external electrical field was analyzed by minimizing the free energy of ferromagnetic and ferroelectric phases with elastic interactions between them. It is shown that the applied electrical field results in either continuous or discontinuous magnetization of magnetic nanorods embedded into the ferroelectric matrix. The experimental results for CoFe2O4-PbTiO3 nanostructure grown on SrTiO3 substrate are used to illustrate the theory.

January 22:
Speaker: Vahid Tarokh, Harvard University - Division of Engineering and Applied Sciences
Title: Bounds on Sparse Representations using Redundant Frames
Abstract: We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N^2) operations, as long as the number of non-zero coefficients in the sparse representation of r is \epsilon N for some 0 \le \epsilon \le 0.5, thus improving on a result of Candes and Tao. We also show that \epsilon \le 0.5 cannot be relaxed without violating uniqueness.
The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.
We also provide numerical results comparing the performance of various sparse representation algorithms using random frames to our bounds.
This is a joint work with Mehmet Akcakaya

January 29: - Joint with the Bio-math Seminar
Speaker: Pilhwa Lee, NYU - Courant Institute of Mathematical Sciences
Title: Immersed Boundary Method with Advection-Electrodiffusion
Abstract: The immersed boundary method is a mathematical and computational framework for problems involving the interaction of a fluid with immersed structures. In the talk, we consider also the role of solutes (possibly charged) and their interactions with membranes. We propose a numerical scheme for the advection-diffusion of solutes in fluid-solute-structure interaction. The transport of solute across possibly moving boundaries is controlled by a chemical barrier along the boundary. Moreover, when the solutes are electrical ions, they generate an electrical potential according to the Poisson equation, and they drift relative to the fluid according to the gradient of the electrical potential. Both explicit and implicit numerical schemes are considered for the advection-electrodiffusion equations. The results show electroneutrality except in space charge layers near membranes, and agree with the Nernst equation for the potential difference across membranes.

February 5: - Joint with the Bio-math Seminar
Speaker: Jian Du, SUNY at Stony Brook
Title: Numerical Study of MHD Effects on Free Surface Liquid Metal Jet with Low Magnetic Reynolds Numbers
Abstract: A numerical algorithm for the study of magnetohydrodynamics (MHD) of free surface flows at low magnetic Reynolds numbers is presented. It employs the method of front tracking for material interfaces, second order Godunov-type hyperbolic solvers, and the Embedded Boundary Method for the elliptic problem in complex domains. The code has been validated through the comparison of numerical simulations of a liquid metal jet in a non-uniform magnetic field with experiments and theory. Simulations of the Muon Collider/Neutrino Factory target will also be discussed, which include mathematical modeling of complex flows undergoing phase transitions.

February 12: - Joint with the Stochastics Seminar
Speaker: Firas Rassoul-Agha, U. of Utah
Title: Almost-sure invariance principle for random walk in random environment
Abstract: Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant.
Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment.
We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for such a process.
This is joint work with Timo Seppalainen (Madison-Wisconsin).

February 26: - Joint with the Stochastics Seminar
Speaker: Martin Wainwright, University of Berkeley - Dept. of Statistics and Electrical Engineering
Title: Sparsity recovery in the high-dimensional and noisy setting: Practical and information-theoretic limitations
Abstract: The problem of recovering the sparsity pattern of an unknown signal arises in various areas of applied mathematics and statistics, including constructive approximation, compressive sensing, and model selection. The standard optimization-theoretic formulation of sparsity recovery involves l_0-constraints, and typically leads to computationally intractable problems. This difficulty motivates the development and analysis of approximate methods; in particular, a great deal of work over the past decade has focused on the use of l_1-relaxations and related convex methods for sparsity recovery.
We consider the high-dimensional and noisy setting, in which one makes n noisy observations of an unknown signal in p dimensions with at most s non-zero entries. Of interest is the number of observations n that are required, as a function of the model dimension p and sparsity index s, to correctly estimate the support of the signal. For a broad class of random Gaussian measurement ensembles, we provide sharp upper and lower bounds on the performance of a computationally efficient method (l_1-constrained quadratic programming), as well as information-theoretic upper and lower bounds on the performance of any method (regardless of its computational efficiency). We discuss connections to other work, and some open problems in this rapidly-growing field.

March 5:
Speaker: Yury Grabovsky, Temple University
Title: Buckling of slender bodies: Universality and link with flip instability
Abstract: Buckling has been understood either as a bifurcation in dimensionally reduced models for rods and plates or exhibited explicitly for 3D non-linearly elastic bodies with simple geometry and constitutive law. One can view buckling as an failure of second variation for to stay positive for 3D slender bodies under compressive loading. The source of that behavior of second variation is the principle of objectivity that is also responsible for flip instability in a purely soft device. One can view buckling as a delayed flip in a mixed device. Buckling occurs when the stabilizing effect of energy convexity and mixed device loading expressed by the Korn constant is overcome by the destabilizing effect of the compressive loading, whose quantitative characteristics will be introduced in this talk. Our theory is largely independent of the precise details of geometry, loading or constitutive anisotropy and non-linearity. As such, it applies to complex geometries.

March 12:
Speaker: Joe Pasciak, U. of Texas A&M
Title: PML and the computation of resonances in open systems.
Abstract: In this talk, I will consider the problem of computing resonances in open systems. I will first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements. We shall see that these both of these steps lead to eigenvalue convergence to the desired resonance values and are free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large and the mesh size is sufficiently small. We illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.

March 26:
Speaker: David George, U. of Utah
Title: Adaptive Shock-Capturing and Well-Balanced Methods for Tsunami Modeling
Abstract: Simulating transoceanic tsunamis at the global scale and modeling inundation at the local coastal scale, presents distinct numerical challenges due to the disparate properties exhibited by these two flow regimes. Although the shallow water equations are typically used to model both of these regimes, numerical methods that are well suited for one regime are often poorly suited for the other. We have developed shock-capturing finite volume methods that are robust and accurate in the local inundation regime. This requires Riemann solvers with special properties, such as nonnegative depth preservation and shoreline capturing. Additionally, by developing Riemann solvers that are well balanced with respect to all smooth steady states, the methods can accurately model transoceanic propagation. We use adaptive mesh refinement so that the methods can be used for transoceanic tsunami propagation and inundation in single global scale simulations. I will describe these difficulties, the algorithms and show some recent results.

April 2:
Speaker: Liping Liu, CalTech
Title: Multiscale Analysis and Modeling Ferromagnetic Shape Memory Composite
Abstract: We calculate the effective properties of a ferromagnetic shape memory (FSM) composite in the cases of the dilute limit and finite volume fraction. The composite consists of identical FSM particles, surrounded by an elastic matrix. The free energy of the FSM particles is computed using the constrained theory of DeSimone and James (2002), where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and magnetostatic terms.
In the case of the dilute limit, by using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. In the case of finite volume fraction, we assume the composite has periodic structure and the embedded FSM particles are much smaller than the overall composite body. Using multiscale methods, we again manage to cast the minimization problem as a quadratic programming problem, provided some special microstructures exist.
The existence problem of these special microstructures is solved in 2D but only partially solved in higher dimensional space. They are constructed as the coincident set of a related free-boundary problem. These special microstructures apparently enjoy many interesting properties with respect to homogenization and energy minimization. In particular, we use them to give new results on a) optimal bounds of the effective moduli of two-phase composites, b) energy-minimizing microstructures; and c) the characterization of the G-closure of two well-ordered conductivity composites.

April 9:
Speaker: Mark Lammers, University of North Carolina at Wilmington
Title: Sigma Delta and Alternate Dual Frames for Reconstruction.
Abstract: We explore reducing errors in digital to analog reconstruction where the original signal has been digitized using one of a class of 1 bit sigma delta algorithms. The underlying structure of the representations is based on frame theory and we will show that the canonical dual is not optimal for reconstructing a signal that has been quantized using these sigma delta algorithms.
For an application we will reconstruct and audio signal and show we can improve the SNR (signal to noise ratio) of the reconstruction by as much as 71% by using an alternate dual.

April 16:
Speaker: Grady Wright, U. of Utah
Title: An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics
Abstract: A gel consists of two-phases, a networked polymer and a fluid solvent. The mechanical and rheological properties of gels can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, gels are important in many biological systems, e.g. gels make up the cytoskeleton and cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. The models of gel dynamics we are considering consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint. The momentum and incompressibility equations present the greatest numerical challenges since i) they involve partial derivatives with variable coefficients that can vary quite significantly throughout the domain (when the phases separate), and ii) their approximate solution requires the "inversion" of a large linear system of equations arising from a finite difference discretization. We discuss an efficient and robust algorithm for solving this system which uses a specially designed multigrid method as a preconditioner for the generalized minimum residual (GMRES) method. To simulate the gel model, we couple this solver with a conservative finite volume method for discretizing the transport equations. Numerical results showing the near linear scalability and robustness of the algorithm are presented.

April 23:
Speaker: Alex Panchenko
Title: G-convergence and homogenization of viscoelastic flows
Abstract: G-convergence is one of the most general tools for analysis of effective behavior of composite materials. In the talk we discuss a possible use of G-convergence for deriving effective equations of materials with moving interfaces. Some general definitions and properties of G-convergence are given first. Then we focus on oscillating test functions as a tool for describing G-limit operators. The method presented in the talk follows up on some ideas contained in the late 1970s papers of Zhikov, Kozlov and Oleinik. Our construction is different from the one used in the classical paper by Murat and Tartar.
The new method works in the situations when the classical method does not. In particular, we discuss a nonlinear evolution of a two-phase incompressible viscoelastic flow with arbitrary disordered microstructure. The effective equations of this flow contain a long memory term not present in the epsilon-problems.

May 10: SPECIAL TIME and DATE, 2:00 - 2:50 PM
Speaker: Avram Sidi, Technion - Department of Computer Science
Title: De Montessus Type Convergence Study for a Vector-Valued Rational Interpolation Procedure
Abstract: Let F(z) be a vector-valued function. Recently, we proposed new vector-valued rational interpolation procedures for F(z). In these procedures, the interpolants R_(p,k)(z) are such that R_(p,k)(z)=U_(p,k)(z)/V_(p,k)(z), where U_(p,k)(z) is a vector-valued polynomial of degree at most p-1 and V_(p,k)(z) is a scalar-valued polynomial of degree k. We first show that R_(p,k)(z) has a determinant representation. We then make use of this representation to present a de Montessus type convergence study [concerning the asymptotic behavior of R_(p,k)(z) as p -> infinity while k is being held fixed] for the case in which F(z) is analytic in a compact set E and meromorphic in a bigger set E' containing E in its interior and the points of interpolation are all in E.

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