October 8 (Wed)
Speaker: Bacim Alali, University of Utah
Title: Multiscale analysis of heterogeneous media in the Peridynamic formulation
Abstract: We present a multiscale method for modeling the dynamics of fiber-reinforced composites using the peridynamic formulation, which is a nonlocal theory of continuum mechanics. The multiscale analysis delivers a multiscale numerical method that captures the dynamics at structural length scales while at the same time is capable of resolving the dynamics at the length scales of the fiber reinforcement.
October 20 (Mon) in LCB 115 (Computer lab)
Speaker: Nelson Beebe, University of Utah
Title: TBA
Abstract: TBA
October 22 (Wed)
Speaker: Masaki Iino, (Student talk) University of Utah
Title: TBA
Abstract: TBA
October 29 (Wed)
Speaker: Graeme Milton, University of Utah
Title: TBA
Abstract: TBA
November 12 (Wed)
Speaker: Andrej Cherkaev, University of Utah
Title: TBA
Abstract: TBA
December 3 (Wed)
Speaker: Guillaume Bal, Columbia University
Title: TBA
Abstract: TBA
August 15 (Friday), LCB 222, 10:30am
Speaker: Peter B. Weichman, British Aerospace
Title: Inverse problems in urban warfare
Abstract: TBA
August 18 (Monday), Location: LCB 219, 4:15pm
Speaker: Alexander Freidin, Institute of Problems in Mechanical Engineering
Russian Academy of Sciences, St. Petersburg, Russia
Title: Equilibrium, stability and kinetics of two-phase deformations
Abstract: We study two-phase deformations formed as a result stress-induced phase transformations and develop approaches aimed to answer the questions:
Given a material and a straining path, when and what two-phase structures can appear?
How a material transforms from one phase state to another?
The consideration is based on the notion of phase transition zones which comes from the equilibrium considerations, the stability analysis procedures and configuration forces expressions.
September 10 (Wed)
Speaker: Fernando Guevara Vasquez, University of Utah
Title: Edge illumination of extended targets
Abstract: We use the singular value decomposition of the array
response matrix to image selectively the edges of extended
reflectors in a homogeneous medium. We show with numerical
simulations in an ultrasound regime, and analytically in the
Fraunhofer diffraction regime, that information about the
edges is contained in the singular vectors for singular
values that are intermediate between the large ones and
zero. Our results confirm well-known experimental
observations on the rank of the response matrix.
October 1 (Wed)
Speaker: Daniel Onofrei, University of Utah
Title: Approximate cloaking for Helmholtz equation in the finite
frequency range. General theory and numerical results
Abstract: In this talk we will discuss about the possibility of
cloaking materials from monochromatic EM guided waves or acoustics waves using
only nonsingular (regular) cloaks. Although perfect cloaking is impossible
using only regular materials, we will describe the procedure of building an
approximate cloak which achieves cloaking within a certain error independent of
the materials to be cloaked. Two central ideas behind our results are, the use
of a suitable nonsingular transformation of variables and the introduction of a
suitable conducting layer around the material to be cloaked in between the
material and the cloak. We will briefly introduce the main ideas and the
analytical results of our work and will be focused on presenting several
numerical results (for the two dimensional case) complimentary to our analysis,
to highlight the role of the conducting layer and the role of the cloak in the
cloaking process, and to show how the error in the approximate cloaking depends
on the conductivity in the layer. In all our numerical results extremely
singular materials (analytically described) to be cloaked will be considered.
We will also present the analytical arguments used to obtain such materials and
will numerically highlight their singular behavior. The numerical
exemplification of the approximate cloak for these materials and a given
incoming plane wave will be presented.
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 (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.
September 6: SPECIAL DATE and TIME, 4:15pm-5:20pm
Speaker: Gregory Gutin,
Royal Holloway, University of London - Department of Computer Science
Title: Worst Case Analysis of Greedy, Max-Regret and Other Heuristics
for Multidimensional Assignment and Traveling Salesman Problems
Abstract: Combinatorial optimization heuristics are often compared with
each other to determine which one performs best by means of worst-case
performance ratio which reflects the quality of returned solution in the worst
case. The domination number is a complement parameter indicating the quality of
the heuristic in hand by determining how many feasible solutions are dominated
by the heuristic solution.
We prove that the Max-Regret heuristic introduced by Balas and Saltzman finds
the unique worst possible solution for some instances of the
s-dimensional (s≥3) assignment problem (s-AP) and the
asymmetric traveling salesman problems (ATSP) of each possible size. It was
proved earlier that Greedy has the same property for ATSP and it's not
difficult to show that Greedy has the same property for s-AP
(s≥2). This means that the domination number of all above mentioned
heuristics (for ATSP and s-AP) is 1.
We show that the Triple Interchange heuristic (for s=3) also introduced
by Balas and Saltzman and two new heuristics (Part and Recursive Opt Matching)
have factorial domination numbers for s-AP (s≥3). ATSP
heuristics of factorial domination number will also be discussed.
The results of preliminary computational experiments with our heuristics will
be shown.
(joint work with B. Goldengorin and J. Huang)
September 10:
Speaker: François Willot, Mechanical Engineering and Applied Mechanics, University of Pennsylvania
Title: Strain localization and effective medium properties
in 2D perfectly-plastic porous materials: the "dilute" limit
Abstract: This work addresses a notoriously difficult problem of nonlinear behavior and infinite contrast between two phases, one of which is a plastic solid phase, and the other one the porosity of the medium. Such problem is of special interest to effective-medium approximations, which typically reach their limits in situations of strong nonlinearity and high contrast between the phases. The aim of this study is to investigate how plastic strain localization manifests itself at the level of the overall effective behavior of the medium in presence of pores, and in particular in the non-trivial limit of small porosity. This question, important to the understanding of ductile damage, is examined both numerically and theoretically, in the special case of two dimensional systems, and with a deformation-theory approach of plasticity. The numerical investigations consist of quasi-exact computations of the stress and strain fields in the voided medium, by means of the Fast Fourier Transform method making use of a particular choice for Green's function. The theoretical approach makes use of exact solutions, which can be obtained in particular cases of a periodic void lattice, as well as of a recent "second-order" nonlinear homogenization approach. The virtues of the latter are evaluated in two steps, first by studying the underlying linear anisotropic homogenization step (an essential ingredient), then by studying the nonlinear step itself. A connection between the strain/stress localization patterns and the macroscopic behavior is shown in the case of a strongly anisotropic linear material. In the nonlinear case, the nature and significance of the singularities, confirmed by FFT computations, are partly elucidated.
September 24:
Speaker: Fernando Guevara Vasquez, University of Utah - Dept. of Mathematics
Title: Electric impedance tomography with resistor networks
Abstract:
Electric impedance tomography consists in finding the conductivity inside a
body from electrical measurements taken at its surface. This is a severely
ill-posed problem: any numerical inversion scheme requires some form of
regularization. We present inversion schemes that address the instability of
the problem by seeking a sparse parametrization of the unknown conductivity.
Specifically, we consider finite volume grids of size determined by the
measurement precision, but where the node locations are to be determined
adaptively. A finite volume discretization can be thought of as a resistor
network, where the resistors are essentially averages of the conductivity over
grid cells. We show that the model reduction problem of finding the smallest
resistor network (of fixed topology) that can predict meaningful measurements
of the Dirichlet-to-Neumann map is uniquely solvable for a broad class of
measurements. We propose a simple inversion method that is based on an
interpretation of the resistors as conductivity averages over grid cells, and
an iterative method that improves such reconstructions by using sensitivity
information on the changes in the resistors due to small changes in the
conductivity. A priori information can also be incorporated to the latter
method.
October 22: CES-CSAFE Seminar (SPECIAL TIME AND LOCATION: 3PM in Warnock Engineering Building 2230)
Speaker: Marsha Berger, New York University, Computer Science Department
Title: Cartesian Cut Cell Methods: Where Do Things Stand?
Abstract: (From the SCI Seminar series)
We discuss some of the steps involved in preparing for and
carrying out a fluid flow simulation in complicated geometry.
Our goal is to automate this process as much as possible to enable
high quality inviscid flow calculations. We use multilevel Cartesian
meshes with irregular cells only in the region intersecting a solid
object. We present some of the technical issues involved in this
approach, including the special discretizations needed to avoid loss of
accuracy and stability at irregular boundary cells, as well as how we
obtain highly scalable parallel performance. This method is in routine
use for aerodynamic calculations in several organizations, including the
NASA Ames Research Center. Many open problems are discussed.
October 29:
Speaker: Jeff Blanchard, University of Utah,
Mathematics Dept.
Title: Composite Dilation Wavelets
Abstract: We will begin by recalling the basic properties of wavelets
including the structure of a multiresolution analysis (MRA). Wavelets are
limited in certain applications due to the rigid geometry of their support
sets. A recent answer to this rigidity introduced by Guo, Labate, Lim, Weiss,
and Wilson is a true generalization of wavelets, Composite Dilation Wavelets.
These affine systems use two sets of dilations, one expanding and one a group
action on R^n. We will discuss how the basic properties of wavelets including
the MRA extend to the composite dilation setting. Via examples, we will discuss
some significant advantages to the composite dilation systems including
non-separable, singly generated, Haar-type wavelets. Time permitting we will
discuss the existence of a very large family of minimally supported frequency
composite dilation wavelets in every dimension.
November 5:
Speaker: Valy Vardeny*, University of Utah - Physics Department
Title: Experimental Studies of Plasmonic Metamaterials
Abstract:
Artificially structured materials, or “metamaterials”, with
properties not present in naturally occurring materials have attracted
significant interest in recent years because their potential to revolutionalize
our understanding of the dielectric function and consequent optical response of
these structures. Three dimensional (3D) metallic photonic crystals, and 2D
periodic and aperiodic arrays of subwavelength apertures on metal films are two
specific examples of such media. The subwavelength nature of the active surface
plasmon polariton (SPP) excitations in such metamaterials, along with strong
field localization open up novel applications in bio-sensing, guided-wave
devices and quantum optics.
Our work has been primarily focused on the fundamental investigation and
development of 2D and 3D plasmonic metamaterials that are active in the
visible, near infrared and terahertz (THz) frequencies. We fabricate 3D
metallo-dielectric photonic crystals based on metal infiltrated opal photonic
crystals, and measure their optical and thermal emission properties. We also
fabricate 2D subwavelength aperture arrays (plasmonic lattices) and use THz
time-domain spectroscopy (THz-TDS) to measure their extraordinary transmission
properties. We demonstrate that aperture periodicity is not crucial for
obtaining strong transmission resonances through these 2D structures, by
measuring the transmission properties of various “designed”
aperture arrays that include quasicrystals and quasicrystal approximates. We
found, however that the thermal emission properties of plasmonic lattices are
not fundamentally different than that of non-perforated metal films, except for
an ‘optical filtering’ effect.
Furthermore, the THz-TDS method that we use allows for a direct measurement of
the THz electric field transmitted through the plasmonic lattices, yielding
both amplitude and phase information. Hence the complete complex dielectric
response of these complex media can be directly measured without resorting to
Kramers-Kronig relation. By treating periodic and aperiodic aperture arrays as
“effective” plasmonic media in the THz beam path, we demonstrate the
ability to engineer the dielectric function of such structures. This may prove
important in understanding the dielectric properties of a broader range of
metamaterials.
* In collaboration with Profs. Efros and Nahata; Drs. Dewkar, Matsui, Pokrovsky
and Kamaev; and Mr. Agrawal.
November 19:
Speaker: Frederic Noo, University of Utah - Utah Center for
Advanced Imaging Research
Title: An excursion into the mathematics of image reconstruction in
single photon emission computed tomography
Abstract:
Single photon emission computed tomography (SPECT) is a particular imaging technique that allows visualization of the distribution of a radio-active tracer in a body in a non-invasive way. In this talk, will review the fundamental equations that relate measurements that can be taken to this distribution, and discuss various ways to recover the distribution from the measurements.
November 28: Joint with the Bio-math seminar, SPECIAL DATE AND TIME (Wednesday at 3:05pm in LCB 215).
Cancelled.
Speaker: Kevin Lin, University of Arizona, Mathematics Dept.
Title: Reliability of coupled oscillators
Abstract:
This talk concerns the reliability of coupled oscillator networks in response
to complex, fluctuating stimuli. Reliability means that repeated presentations
of a stimulus elicit essentially identical responses regardless of the system's
state at the onset of the input. This work is motivated by basic questions from
neuroscience, where the reliability of a network is relevant to how information
may be encoded and transmitted. I will show how the question of reliability can
be precisely formulated in the framework of random dynamical systems theory,
and review the well-known fact that single phase oscillators are reliable. I
will then show that unreliability can occur even in a 2-oscillator system, and
propose a geometric mechanism for the observed phenomena. The talk will
conclude with some observations concerning larger networks, including a natural
condition which precludes unreliability. No prior knowledge of random dynamical
systems theory is assumed. This is joint work with Eric Shea-Brown and Lai-Sang
Young.
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.
August 30: SPECIAL TIME, DATE, and LOCATION, 2:15 - 3:05 PM, LCB 225
Speaker: Amy Shen, Washington University in St. Louis - Dept. of Mechanical and Aerospace Engineering
Title: Hydrodynamics of complex fluids at small length-scales
Abstract: Understanding fluid transport and interfacial phenomena of
complex fluids at small length-scales is crucial to understanding how to design
and exploit of micro- and nano-fluidic devices. I will present two
examples. The first studies evaporation driven self-assembly to
synthesize nanoporous thin films. A combination of experimental
measurement and modeling using lubrication theory shows how
self-assembly influences coating film thickness. The second example
studies how length-scale and fluid elasticity affect droplet pinch-off
of "simple" polymeric liquids in microfluidic
environments. Boger fluids (viscoelastic liquids with nearly constant
shear viscosity) are pumped into microchannels and pinched off to form
droplets in an immiscible oil
phase. We find a power law relation between the dimensionless
capillary pinch-off time and the so-called elasticity number, E, of the
fluid. Theoretical models that neglect the extensional viscosity of the
fluid become increasingly more inaccurate as the fluid elasticity
increases.
September 5: SPECIAL TIME, DATE, and LOCATION, 3:05 - 3:55 PM,
LCB 115
Joint with the Approximation Theory Seminar
Speaker: Peter Alfeld, University of Utah
Title: The Bernstein Bézier Form of a Multivariate Polynomial
Abstract: The Bernstein-Bézier form (or just B-form) of a polynomial
is a highly successful and widely used way of representing polynomials,
particularly polynomials in more than one variable. Its power stems
from the fact that algebraic issue, such as two polynomials joining
smoothly, can be studied and interpreted geometrically. There is also
a close geometric connection between the coefficients of a polynomial
and the shape of its graph. In this talk I will define the B-form of a
polynomial and discuss some of its properties. This will serve as the
foundation for several future talks this semester. The talk will
include computer demonstrations.
September 11: - Joint with the Stochastics Seminar
Speaker: Eulalia Nualart, University of Paris 13
Title: Potential theory for non-linear stochastic heat equations
Abstract: In this talk we develop potential theory for a system of
non-linear stochastic
heat equations in spatial dimension one and driven by a space-time white
noise.
In particular, we prove upper and lower bounds on hitting probabilities of
the process which is solution
of this system of equations, in terms of respectively Hausdorff measure and
Newtonian capacity.
These estimates make it possible to discuss polarity for points and to
compute the Hausdorff dimension
of the range and the level sets of this process. In order to prove the
hitting probabilities estimates,
we need to establish Gaussian type bounds for the bivariate density of the
process in order to quantify its
degenerance. For this, we use techniques of Malliavin calculus.
September 18:
Speaker: Joel Tropp, University of Michigan at Ann Arbor
Title: Sparse solutions to underdetermined linear systems
Abstract: A central problem in electrical engineering,
statistics, and applied mathematics is
to solve ill-conditioned systems of linear equations. Basic linear
algebra forbids
this possibility in the general case. But a recent strand of
research has
established that certain ill-conditioned systems can be solved
robustly with
efficient algorithms, provided that the solution is sparse (i.e.,
has many zero
entries). This talk describes a popular method, called l1
minimization or Basis
Pursuit, for finding sparse solutions to linear systems. It details
situations where
the algorithm is guaranteed to succeed. In particular, it describes
some new work on
the case where the matrix is deterministic and the sparsity pattern
is random. These
results are currently the strongest available for general linear
systems.
September 25: - Joint with the Approximation Theory
Seminar
Speaker: Yuliya Babenko, Sam Houston State University
Title: On asymptotically optimal methods of adaptive spline interpolation
Abstract: In this talk we shall present the exact asymptotics of the optimal error in the weighted $L_p$-norm, $1\leq p \leq \infty$, of linear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. The connections with the problem of approximating the convex bodies by polytopes and the problem of adaptive mesh generation for finite element methods will also be discussed.
We shall present review of existing results as well as a series of new ones.
Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation. Similar results are obtained for some other classes of splines. We shall also discuss the analogous multivariate results as well.
October 2:
Speaker: Andrej Cherkaev, University of Utah
Title: New bounds for multiphase conducting composites
Abstract: New bounds for effective properties tensors of multimaterial composites are
suggested. These bounds complement the translation bounds or
Hashin-Shtikman bounds. We show that the bounds are exact for
three-material composites and determine optimal microstructurs of them.
The bounds are obtained using the theory of "localized polyconvexity" which
will be also discussed.
October 9: SPECIAL TIME and LOCATION, 12:55 - 1:45 PM, LCB 225
Joint with the Stochastics Seminar
Speaker: Pierre Seppecher, University of Toulon
Title: A closed notion of locality for Dirichlet forms in the one dimensional case
Abstract: If the notion of locality is well known in the case of regular Dirichlet
form, it is is not straightforward to extend it to non-regular forms.
We compare different possible definitions and characterize a notion of
locality which is closed with respect to Mosco or $\Gamma$-convergence.
This enable us to characterize the closure of the set of diffusion
functionals in the one-dimensional case.
October 16:
Speaker: Graeme Milton, University of Utah
Title: Cloaking: a New Phenomena in Electromagnetism and Elasticity
Abstract: Since my talk last semester, there have been quite a number of
developments with regards to the theory of cloaking (making an object
invisible). Not only developments with respect to cloaking associated
with superlenses, as I had discussed, but also with proposals
by Pendry, Schurig and Smith and Leonhardt, for designing a shield
which cloaks objects to electromagnetic waves. This work is related
to the earlier work of Greenleaf, Lassas and Uhlmann, on cloaking
for conductivity. Here we will review these developments and also
discuss how cloaking might be extended to elasticity using these
ideas. This requires new materials, in particular materials with
anisotropic density. We show how such materials can be made.
October 23:
Speaker: Jorge Balbas, University of
Michigan at Ann Arbor - Dept. of Mathematics
Title: Non-oscillatory Central Schemes for One-dimensional Shallow Water Flows along
Channels with Non-uniform Rectangular Cross-sections
Abstract: We present a new high-resolution, non-oscillatory semi-discrete central
scheme for one-dimensional shallow water flows along channels with non-uniform rectangular
cross sections. The scheme extends existing central semi-discrete schemes for hyperbolic
conservation laws by incorporating a discretization of the source terms appearing in shallow
water equations so that nonlinear fluxes are balanced
for steady-state solutions. We also incorporate exact information in the polynomial
reconstruction of the wet area, improving the control of spurious oscillations. Along with
the scheme, we present a systematic approach to calculate exact steady-state solutions for the
balance law. This allows us to validate the scheme by comparing the approximate numerical
solutions to the exact ones.
October 30:
Speaker: Adam Oberman, Simon Fraser University - Dept. of Mathematics
Title: Fully nonlinear elliptic PDEs: models, applications, and solution methods.
Abstract: This will be an accessible talk about modeling using fully nonlinear elliptic
PDEs.
Modern applications of these PDEs are to Image Processing and Math Finance.
As well as the Level Set Method for curve evolution, Optimal Control and
Stochastic Control.
I'll discuss some interesting models, overview the relevant theory, and then
show how to solve these equations.
Examples will include: level set motion by mean curvature, the convex hull,
the infinity Laplacian, as well as examples from math finance and control
theory.
We will present results which allow schemes to be built for a wide class of
equations.
November 6:
Speaker: Jonathan Kaplan, Stanford University - Dept. of Mathematics
Title: The Morphlet Transform: A Multiscale Transform for Diffeomorphisms
Abstract: Diffeomorphisms are a classical tool and object of study in
theoretical mathematics. Recently, there has been an increase in
the use and study of diffeomorphisms in applied mathematics. In
particular, diffeomorphisms have appeared as a new and potent tool
in image analysis. There is a growing interest in understanding
computationally efficient mechanisms for representing and
manipulating diffeomorphisms. Inspired by the success of wavelets
in signal processing, we describe a multiscale transform acting on
diffeomorphisms. This transform is defined on dyadic samples and
is nonlinear. Its design draws from the theory of interpolating
wavelet transforms and nonlinear subdivision schemes. We call
this transform the morphlet transform.
November 13:
Speaker: Coralia Cartis, Rutherford Appleton Laboratory - Computational Science & Engineering Dept.
Title: Some challenges in interior point methods for linear programming
Abstract: Through the depth of their theory and the span of their successful applications,
interior point methods have sparked a veritable "revolution" in convex
optimization. Now, fifteen years after their landmark discovery, interior point
methods have become highly successful at solving (very) large-scale linear
programming problems, with millions of variables and constraints not uncommon.
Nonetheless, some important questions at the interface of theory and practice
remain open and I will address three such topics in this talk. In particular, I
will present a new way of initializing these algorithms which overcomes the
fundamental assumptions underlying interior point methods theory that require the
set of admissible solutions to be full-dimensional and that are rarely satisfied by
real-life problems (this is joint work with Nick Gould, Oxford University).
Furthermore, addressing the lack of theoretical reliability of the interior point
algorithm implemented in most commercial and public software, I show on an example
what may go wrong and then describe a theoretically reliable alternative. As
interior point methods have made linear programming solvable in polynomial time,
complexity is a crucial aspect of this area. We set up a new general framework in
which we perform such a complexity analysis, that attempts to be more practical and
insightful than existing, highly-constructive, techniques by employing stiffness
analysis of vector fields, a concept traditionally associated with ordinary differential
equations (this is joint work with Raphael Hauser, Oxford University).
November 14: SPECIAL TIME, DATE, and LOCATION, 3:00 - 4:00 PM LCB 215
Special Seminar - Joint with Bio-Math
Speaker: Doron Levy, Stanford University - Mathematics Dept.
Title: Modeling the Dynamics of the Immune Response to Chronic Myelogenous Leukemia
Abstract:
Chronic Mylogenous Leukemia (CML) is a blood cancer with a common acquired
genetic defect resulting in the overproduction of malformed white blood
cells. The cause of CML is an acquired genetic abnormality in
hematopoietic stem cells in which a reciprocal translocation between
chromosomes 9 and 22 occurs. It is this abnormality that leads to
dysfunctional regulation of cell growth and survival, and consequently to
cancer. Treatment and control of CML underwent a dramatic change with the
introduction of the new drug, Gleevec, which was shown to be an effective
treatment available to nearly all CML patient. Nevertheless, by now it
is widely agreed that Gleevec does not represent a true cure for CML,
with many patients beginning to relapse despite of continued therapy.
The only known treatment that can potentially cure CML is a bone-marrow
(or stem-cell) transplant.
In this talk we will describe our recent works in modeling the interaction
between the immune system and cancer cells in CML patients. One model
follows this dynamics after a stem-cell transplant. A second model follows
the immune-cancer dynamics in patients treated with Gleevec. Related
mathematical questions and possible exciting applications of the models will
be discussed. This is a joint work with Peter Kim and Peter Lee
(Hematology, Stanford Medical School).
November 17: SPECIAL TIME, DATE, and LOCATION, 3:30 - 4:20 PM LCB 121
Special Seminar - Joint with the Approximation Theory Seminar
Speaker: Frank Stenger, U. of Utah - Computer Science Dept.
Title: SINC-PACK Enables Separation of Variables
Abstract:
This talk is mainly for mathematicians. It consists of a "proof-part"
of Stenger's SINC-PACK computer package (an approx. 400-page tutorial
+ about 250 Matlab programs) that one can always achieve separation of
variables when solving linear elliptic, parabolic, and hyperbolic PDE
(partial differential equations) via use of Sinc methods.
Some examples illustrating computer solutions via Sinc-Pack will
nevertheless be given in the talk. For example, in one dimension,
Sinc-Pack enables the following, over finite, semi--infinite, infinite
intervals or arcs: interpolation, differentiation, definite and
indefinite integration, definite and indefinite convolution, Hilbert
and Cauchy transforms, inversion of Laplace transforms, solution of
ordinary differential equation initial value problems, and solution of
convolution-type integral equations. The methods of the package are
especially effective for problems with (known or unknown - type)
singularities, for problems over infinite regions, and for PDE
problems.
In more than one dimension, the package enables solution of linear and
nonlinear elliptic, hyperbolic, and parabolic partial differential
equations, as well as integral equations and conformal map problems,
in relatively short programs that use the above one-dimensional
methods. The regions for these problems can be curvilinear, finite,
or infinite. Solutions are uniformly accurate, and the rates of
convergence of the programs of SINC-PACK are exponential.
In Vol 1. of their 1953 text, Morse and Feshbach prove for the case of
3-dimensional Poisson and Helmholtz PDE that separation of variables
is possible for essentially 13 different types of coordinate systems.
A few of these (rectangular, cylindrical, spherical) are taught in our
undergraduate engineering-math courses. We prove in the talk that one
can ALWAYS achieve separation of variables via use of Sinc-Pack, under
the assumption that calculus is used to model the PDE.
November 20: - Joint with the Approximation Theory
Seminar
Speaker: Tatyana
Sorokina, University of Georgia
Title: Quasi-Interpolation by Multivariate Piecewise Polynomials.
Abstract: Quasi-interpolants provide an alternative to finite elements
in multivariate approximation. While there are reliable tools for studying
classical finite elements, there is no theory of quasi-interpolation in
several variables. We will discuss some theoretical aspects of
quasi-interpolation, consider explicit bivariate and trivariate schemes,
and state open problems.
November 27:
Speaker: Paul Fife, University of Utah
Title: The structure of turbulent flow near boundaries
Abstract:
The problem of deriving key features of steady turbulent flow adjacent
to a wall has drawn the attention of some of the most noted fluid
dynamicists of all time. Standard examples of such features are found
in the mean velocity profiles of turbulent flow in channels, pipes or
boundary layers. Possibly the best known of the elementary
theoretical efforts along this line, and certainly the simplest, is
the argument (obtained independently) by Izakson (1937) and Millikan
(1939) regarding the mean velocity profile. They showed that if an
inner scaling and an outer scaling for the profile are valid near the
wall and near the center of the flow (or the edge of the boundary
layer), respectively, and if there is an overlap region where both
scalings are valid, then the profile must be logarithmic in that
common region. That piece of theory has been used and expanded upon by
innumerable authors for over 60 years, and at the present time is
still rightly enjoying popularity.
The main foci of the talk will be on (i) a careful examination of the
Izakson-Millikan argument, and (ii) an account of a newer approach to
gaining theoretical understanding of the mean velocity and Reynolds
stress profiles, due to Klewicki, McMurtry, Metzger, Wei and
myself. It has similar goals but entirely different methods. The
question will be how, and how well, do these arguments supply insight
into the structure of the mean flow profiles? Answering the question
WHY? is even more important than WHAT?
December 4:
Speaker: Patrick J. Wolfe, Harvard University - Department of Statistics
Title: Time-Frequency Representations and Statistical Models for Speech:
Exploring the Space of Acoustic Waveform Variation
Abstract: Time-frequency representations are ubiquitous in audio signal processing,
their use being motivated by both auditory physiology and the mathematics of
Fourier analysis. Indeed, information-carrying natural sound signals can
often be meaningfully represented as a superposition of translated, modulated
versions of a simple window function exhibiting good time-frequency
concentration. In combination with statistical models formulated in the
space of time-frequency coefficients, such an approach provides a principled
way of decomposing sounds into their constituent parts, as well as an
effective means of exploiting the local correlation present in the
time-frequency structure of natural sound signals such as speech. In
addition to applications involving the reconstruction of noisy and missing
audio data, this talk will describe the ways in which generative statistical
models provide a means of exploring the space of acoustic waveform variation,
and how in doing so they point toward a new way forward in a variety of
speech processing and discrimination tasks.