epshteyn (at) math.utah.edu)August 22 (Welcome Back and Group Photo!)
Speaker: Davit
Harutyunyan, Department of Mathematics, The University of Utah
Title: Quantitative Wulff and Brunn-Minkowski inequalities for convex sets
Abstract: In this lecture we revisit the anisotropic isoperimetric (Wulff) and the
Brunn-Minkowski inequalities for convex sets. The best know constant
C(n)=Cn^8.5 depending on the space dimension n in both inequalities is due
to Figalli, Maggi and Pratelli, 2010. We improve that constant to Cn^6 for
convex sets and even better in some cases. We also conjecture, that the
best constant in both inequalities must be of the form Cn^2, i.e.,
quadratic in n. The tools are the Brenier's mapping from the theory of
optimal mass transportation combined with new sharp geometric-arithmetic
mean and some algebraic inequalities plus a trace estimate by Figalli,
Maggi and Pratelli.
September 26
Speaker: Jared Whitehead,
Department of Mathematics, BYU
Title: Variations in the heat transport of Rayleigh Benard convection
Abstract: Rayleigh Benard convection is the canonical system where a fluid
is heated from below and/or cooled from above, yielding an unstably driven
system. When this temperature difference is sufficiently large, buoyancy
effects induce convective motion within the fluid. As this driving force
increases, the flow becomes turbulent. The fundamental question asked of
this system is how the volume averaged heat transport depends on the
driving force and material properties of the fluid. After reviewing the
sometimes contradictory experimental evidence, we consider how different
boundary conditions, heat sources, and variations in the kinematic
properties of an incompressible fluid affect the heat transport in the
turbulent, convective regime using rigorous upper bounds derived via
variational techniques.
October 17
Speaker: Alexander Kurganov,
Department of Mathematics, Tulane University
Title: CENTRAL-UPWIND SCHEMES FOR SHALLOW WATER MODELS
Abstract: In the first part of the talk, I will describe a general framework for
designing finite-volume methods (both upwind and central) for
hyperbolic systems of conservation laws. I will focus on
Riemann-problem-solver-free non-oscillatory central schemes and, in
particular, on central-upwind schemes that belong to the class of
central schemes, but has some upwind features that help to reduce the
amount of numerical diffusion typically present in staggered central
schemes such as, for example, the first-order Lax-Friedrichs and
second-order Nessyahu-Tadmor scheme.
In the second part of the talk, I will discuss how central-upwind
schemes can be extended to hyperbolic systems of balance laws, such as
the Saint-Venant system and related shallow water models. The main
difficulty in this extension is preserving a delicate balance between
the flux and source terms. This is especially important in many
practical situations, in which the solutions to be captured are
(relatively) small perturbations of steady-state solutions. The other
crucial point is preserving positivity of the computed water depth
(and/or other quantities, which are supposed to remain nonnegative). I
will present a general approach of designing well-balanced positivity
preserving central-upwind schemes and illustrate their performance on a
number of shallow water models.
October 24
Speaker: Noa Kraitzman, Department of Mathematics, The University of Utah
Title: Bifurcation and Competitive Evolution of Network Morphologies in
the strong Functionalized Cahn-Hilliard (FCH) Free Energy
Abstract: The FCH is a higher-order free energy for blends of amphiphillic polymers and solvent which balances solvation energy of ionic groups against elastic energy of the underlying polymer backbone. Its gradient flows describe the formation of solvent network structures which are essential to ionic conduction in polymer membranes. The FCH possesses stable, coexisting network morphologies and we characterize their geometric evolution, bifurcation and competition through a center-stable manifold reduction which encompasses a broad class of coexisting network morphologies. The stability of the different networks is characterized by the meandering and pearling modes associated to the linearized system. For the $H^{-1}$ gradient flow of the FCH energy, using functional analysis and asymptotic methods, we drive a sharp-interface geometric motion which couples the flow of co-dimension 1 and 2 network morphologies, through the far-field chemical potential. In particular, we derive expressions for the pearling and meander eigenvalues for a class of far-from-self-intersection co- dimension 1 and 2 networks, and show that the linearization is uniformly elliptic off of the associated center stable space.
October 31
Speaker: Michael Ryvkin, The Iby
and Aladar Fleischman Faculty of Engineering, Tel Aviv University
Title: ANALYSIS OF NON-PERIODIC STRESS STATE IN PERIODIC MATERIALS.
Applications to fracture and optimization.
Abstract: Many man-made materials have a periodic microstructure, periodic materials are widely met also in nature. Study of overall elastic properties of such materials and their optimization is a well-developed topic. The corresponding problems are characterized by a periodic stress state, however, in many cases of interest the stress state in periodic material is non-periodic. The non-periodicity can result from a non-periodic applied loading, from presence of cracks, inclusions and other flaws, and from the finite dimensions of the sample to be considered.
In these cases a direct numerical simulation is computationally expensive due to a large number of degrees of freedom to be involved, but reducing the analysis domain to a single repetitive cell is not straightforward. This goal is achieved by applying the discrete Fourier transform, casted as the representative cell method. As a result, one has to resolve a number of representative(repetitive) cell problems in the transforms space and can obtain the sought elastic field by the inverse transformation. The important feature of these problems is that they are independent and, consequently, can be treated by the use of parallel computing. It is shown how to plug-in the method into an efficient multiscale analysis scheme for arbitrary shaped sample of periodic material.
The suggested approach is employed for the fracture analysis of beam lattices: two-dimensional honeycombs and spatial open cell Kelvin foam, both cracks nucleation and propagation problems are addressed. Solid periodically voided and composite materials with flaws are considered as well, the optimal parameter combinations maximizing the fracture toughness are determined.
November 7
Speaker: Orly
Alter, Departments of Bioengineering and Human Genetics, The University of Utah
Title: Cancer Diagnostics and Prognostics from Comparative Spectral Decompositions of
Patient-Matched Genomic Profiles
Abstract: I will, first, briefly review our matrix and tensor modeling of large-scale
molecular biological data, which, as we demonstrated, can be used to correctly
predict previously unknown physical, cellular, and evolutionary mechanisms that
govern the activity of DNA and RNA. Second, I will describe our recent generalized
singular value decomposition (GSVD) and tensor GSVD comparisons of the genomes of
tumor and normal cells from the same sets of astrocytoma brain and, separately,
ovarian cancer patients, which uncovered patterns of DNA copy-number alterations
that are correlated with a patient's survival and response to treatment. Third, I
will present our higher-order GSVD, the only mathematical framework that can create
a single coherent model from, i.e., simultaneously find similarities and
dissimilarities across multiple two-dimensional datasets, by extending the GSVD from
two to more than two matrices.
November 14
Speaker: Lajos Horvath, Department of Mathematics, The University of Utah
Title: Eigenvalue analysis of large dimensional matrices
Abstract: Testing for stability in linear factor models has become an important topic in both the statistics
and econometrics research communities. The available methodologies address testing for changes in
the mean/linear trend, or testing for breaks in the covariance structure by checking for the constancy
of common factor loadings. In such cases when an external shock induces a change to the stochas
tic structure of high dimensional data, it is unclear whether the change would be reflected in the
mean, the covariance structure, or both. We develop a test for structural stability of linear factor
models that is based on monitoring for changes in the largest eigenvalue of the sample covariance
matrix. The asymptotic distribu tion of the proposed test statistic is established under the null
hypothesis that the mean and covariance structure of the cross sectional units remain stable during
the observation period. We show that the test is consistent assuming common breaks in the mean
or factor loadings. These results are investigated by means of a Monte Carlo simulation study, and
their usefulness is demonstrated with an application to U.S. treasury yield curve data, in which some
interesting features of the 2007-2008 subprime crisis are illuminated.
Joint work with Gregory Rice.
November 21. Note starting time is 4:15pm.
Speaker: Vianey Villamizar,
Department of Mathematics, BYU
Title: Exact Local Absorbing Boundary Conditions for Acoustic Waves in terms of
Farfield Expansions
Abstract: We devise a new high order local absorbing boundary condition (ABC) for radiating
problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary
shape. By introducing an artificial boundary $S$ enclosing the scatterer, the
original unbounded domain $\Omega$ is decomposed into a bounded computational domain
$\Omega^-$ and an exterior unbounded domain $\Omega^+$. Then, we define interface
conditions at the artificial boundary $S$ , from truncated versions of the
well-known Wilcox and Karp farfield expansion representations of the exact solution
in the exterior region $\Omega^+$. As a result, we obtain a new local absorbing
boundary condition (ABC) for a bounded problem on $\Omega^-$, which effectively
accounts for the outgoing behavior of the scattered field. Contrary to the low order
absorbing conditions previously defined, the order of the error induced by this ABC
can easily match the order of the numerical method in $\Omega^-$. We accomplish this
by simply adding as many terms as needed to the truncated farfield expansions of
Wilcox or Karp. The convergence of these expansions guarantees that the order of
approximation of the new ABC can be increased arbitrarily without having to enlarge
the radius of the artificial boundary. We include numerical results in two and three
dimensions which demonstrate the improved accuracy and simplicity of this new
formulation when compared to other absorbing boundary conditions.
Collaborators: Sebastian Acosta and Blake Dastrup
epshteyn (at) math.utah.edu).
Past lectures: Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.