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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