epshteyn (at) math.utah.edu)January 6
Speaker: Vladislav Babenko, Dnepropetrovsk National University, Institute of Applied Mathematics and Mechanics of NAS of Ukraine
Title: The optimal recovery of integrals of
set-valued functions
Abstract: Approximate calculation of integrals of set-valued functions is important
for many branches of mathematics. For number-valued functions theory of
quadrature and cubature formulae is well developed. At the same time we
know only 3 papers devoted to optimization of approximate calculation of
integrals of set-valued functions.
We systematically consider problems of optimal recovery of integrals of
set-valued functions from various classes using precise information and
information with an error. In particular, we will consider classes of
functions monotone with respect to inclusion and classes of functions
having prescribed majorant of modulus of continuity.
January 27
Speaker: Tom Lyche,
CMA and Department of Informatics, University of Oslo
Title: Simplex Splines
Abstract: Surfaces defined over triangulations have widespread application
in many areas ranging from finite element analysis and physics and
engineering applications to the entertainment industry. For many of these
applications piecewise linear surfaces do not offer sufficient smoothness.
To obtain $C^1$ smoothness, one must either use quintic polynomials with 21
degrees of freedom over each triangle or use lower degree macroelements
that subdivide each triangle into a number of subtriangles. Thus far, the
second approach has largely been based on using the Bernstein-B'ezier
basis on each subtriangle, manually enforcing the smoothness internal to
each triangle and solving the resulting constrained system.
After giving an overview of Simplex splines we consider the S-spline
basis, a B-spline-like basis, over a single macroelement known as the
Powell-Sabin 12-split.
Internal to the macroelement, each of the 12 basis elements is $C^1$ and
the basis is unconditionally stable independent of the shape of the
triangle. Analogous results for analytical and shape properties, so
inextricably intertwined in the B-spline/Bézier formulation of surfaces,
are shown for the S-spline basis.
These results are based on:
E. Cohen, T. Lyche, and R. F. Riesenfeld, A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines, Math. Comp. , Vol. 82, pp.1667-1707, (2013).
February 3 (Special Applied Math Seminar)
Speaker: Ting Zhou, Department of Mathematics, MIT
Title: ON GLOBAL UNIQUENESS FOR AN IBVP FOR THE
TIME-HARMONIC MAXWELL EQUATIONS
Abstract: I will present a uniqueness result for an inverse boundary value problem (IBVP)
for time harmonic Maxwell's equations. We assume that the unknown electromagnetic
properties of the medium, namely the magnetic permeability, the electric permittivity
and the conductivity, are described by continuously di erentiable functions. The IBVP is
a nonlinear problem to determine these parameters using the boundary measurements of
the electromagnetic elds. The key ingredient in proving the uniqueness is the complex
geometrical optics (CGO) solutions.
This is a joint work with Pedro Caro.
February 7 (Special Applied Math Seminar), Note Room LCB 215
Speaker: Braxton Osting, Mathematics Department, UCLA
Title: Extremal Eigenvalue Problems in Geometry and Optics
Abstract:Since Lord Rayleigh conjectured that the disk should minimize the first eigenvalue of the Laplace-Dirichlet operator among all shapes of
equal area more than a century ago, extremal eigenvalue problems have been
an active research topic. In this talk, I'll demonstrate how extremal
eigenvalue problems can arise in geometry and optics and present some
recent analytical and computational results in these areas.
February 21
Speaker: Tamara Sukacheva, Novgorod State
University, Russia
Title: SOBOLEV TYPE EQUATIONS
Abstract:A theory of the solvability of semi-linear non autonomous
sobolev type equations will be presented . On the base of degenerative
resolving semi-groups of operators the uniqueness of the solution of the
Cauchy problem (which is a quasi-stationary trajectory) for these equations
will be proved. The phase space of this problem will be described. The
theory will be illustrated by some models of hydrodynamics.
This is a joint work with prof. G.Sviridyuk, South Ural State University, Russia.
March 3
Speaker: Tomasz Jan Petelenz, Department of Bioengineering, University of Utah
Title: Non-invasive physiological monitoring of cardiac output.
Abstract: Non-invasive physiological measurements are becoming increasingly important in modern healthcare. However, despite the advancements of technology, there are still physiological variables, such as cardiac output that are very difficult, or even impossible to measure non-invasively, especially in out-of hospital environment. Cardiac output is the result of the heart function and in healthy individuals varies in response to the body' demand for oxygen. In severe trauma associated with hemorrhage, such as in injured soldiers on the battlefield, cardiac output could provide critical life-saving diagnostic information and would be critical to rapid assessment and monitoring of the casualty. This presentation will discuss a new cardiac output measurement method using high frequency electromagnetic fields, instrumentation, obtained experimental results, as well as the remaining theoretical and computational (mathematical) challenges that must be solved before a truly portable non-invasive cardiac output measurement system can be developed.
March 17
Speaker: Mikyoung Lim, Department of Mathematical Sciences, KAIST
Title: Characterization and computation of the solution to the conductivity equation in the presence of adjacent inclusions
Abstract: When inclusions with extreme conductivity (insulator or perfect
conductor) are closely located, the gradient of the solution to the
conductivity equation can be arbitrarily large. And computation of the
gradient is extremely challenging due to its nature of blow-up in the
narrow region in between inclusions. In this talk we characterize
explicitly the singular term of the solution when two circular
inclusions with constant conductivities are adjacent. Moreover, we
show through numerical computations that the characterization of the
singular term can be used efficiently for computation of the gradient
in the presence adjacent inclusions.
March 21
Speaker: Robert Palais, Department of Mathematics, Utah Valley University and Pathology Department, University of Utah
Title: Counting, Copies
Abstract: DNA copy number variation is associated with a variety of diseases.
Some involve additional copies of entire chromosomes (e.g. Trisomy 13, 18,
21) while some cancers involve large heterozygous deletions. We will
discuss mathematical contributions to two experimental approaches for
quantifying copy number, one using high resolution melting, and another
using digital PCR. We will also show some recent examples in mathematical
visualization in teaching.
March 24
Speaker: Peter Alfeld, Department of Mathematics, University of Utah
Title: Multivariate Splines
Abstract: Splines are piecewise polynomial functions. They come in two flavors:
univariate (one independent variable), and multivariate (several
independent variables). Univariate splines are used ubiquitously and
routinely in numerical analysis for many purposes, including
approximation of data and functions, numerical integration, solution
of differential equations, designs of curves, interpolation and
approximation. Multivariate splines are much more complicated, their
applicability is more limited, and they exhibit more diversity than
univariate splines. I will focus on piecewise polynomial splines
defined on triangulations and tetrahedral partitions,
and discuss issues, techniques, and open problems.
March 31
Speaker: Jingyi Zhu, Department of Mathematics, University of Utah
Title: Static Hedging of Options with Nearby Contracts
Abstract: The Black-Scholes model for pricing options is based on a dynamic hedging argument,
in which the seller of an option can hedge its risk by dynamically adjusting a
portfolio consisting of the underlying stock and cash. In practice this leads to
transaction costs that cannot always be ignored. Static hedging strategies explore
the relations of options with different contract parameters such as the strike price
and expiration, and they can simplify the management of the portfolio, and finally
they are often model independent. We propose a strategy that is based on finite
difference approximations of the Dupire equation which is a dual equation of the
Black-Scholes PDE. Applying the strategy to S&P 500 index options in a historical
test also shows that we can form many maturity-strike schemes from the available
option contracts that outperform delta hedging with daily
rebalancing.
Joint work with Liuren Wu.
April 7 (Student Talk)
Speaker: Jason Albright, Department of Mathematics, University of Utah
Title: High Order Accurate Difference Potentials Methods for Parabolic Interface Problems
Abstract: The Difference Potentials Method (DPM) was originally introduced in 1969 by Viktor
Ryaben'kii and developed extensively by him and his students. DPM can be viewed as a
discrete analog to the method of generalized Calderón potentials and Calderón
boundary equations with projections. Recently, the Difference Potentials approach
was extended to the spatial discretization of parabolic problems. In this talk, I
will present the development of high-order accurate DPM for variable coefficient
parabolic problems with interfaces. The accuracy of the obtained schemes will be discussed and then illustrated numerically
with 1-D examples. Although, the current numerical results for the aforementioned
high-order accurate methods are illustrated in 1-D settings, the developed
methodology is general and will be extended to variable coefficient parabolic models
in domains with complex geometry in 2-D and 3-D, as well as to the design of the
domain decomposition approaches and parallel algorithms.
This is joint work with Y. Epshteyn and K. R. Steffen.
April 14
Speaker: Stewart Ethier, Department of Mathematics, University of Utah
Title: An optimization problem involving a three-person game
Abstract: We consider the classic game baccara banque (a.k.a. baccara à deux
tableaux), a three-person zero-sum game parameterized by θ ∈ (0,1).
An incomplete study of the game by Downton and Lockwood (1976) argued that the Nash
equilibrium is of only academic interest because “it implies an attitude to
the game by all three participants, which is unlikely to be realized in
practice.” Their preferred alternative is what we call the independent
cooperative equilibrium. But this solution exists only for certain θ. A
third solution, which we call the dependent cooperative equilibrium, always
exists and may be preferable in some respects. The optimization problem of
evaluating these three equilibria will be discussed.
This is joint work with Jiyeon Lee, who will be visiting the department in 2014–15.
April 21
Speaker: Frank Stegner, School of Computing, University of Utah
Title: Wiener - Hopf
Some History and Some New Results
Abstract: TBA
June 16
Speaker: Gal Shmuel, Divison of Engineering and Applied Science, California Institute of Technology
Title: Wavelet Analysis of Microscale Strains
Abstract: Recent experiments and numerical simulations provide
numerous observations on the microstructure and deformation of
polycrystals. These show how confined bands of deformation percolate
in a complex way across various grains. Such information is
represented as samples on grids, and, in turn, creates huge data
sets. The extensive size of data in this form renders identifying key
features difficult, and the cost of digital storage expensive. To
represent, analyze, and predict strain fields with localized features,
we use wavelets: multiresolution functions, which are localized both
in frequency and real domains. By way of example, we focus on
pseudo-elastic polycrystals, capable of recovering strains beyond an
apparent elastic limit. I will show how wavelets efficiently represent
experimental and simulated strains of Ni-Ti, while reducing data size
by two orders of magnitude. More importantly, I will show how the
compact wavelet representation captures the essential physics
within. Finally, I will discuss how to use insights gained to improve
specific experimental and computational methods.
July 16, Joint Math Biology/Applied Math Seminar,
Note Time 3:05pm, Room LCB 219
Speaker: Mark Allen, University of Texas at Austin
Title: The two-phase parabolic Signorini problem.
Abstract: In this talk we will review some obstacle-type problems and their applications.
Particular focus will be given to the thin obstacle problem which in some
instances is the same as the so-called Signorini problem which has applications in
biology and finance. We will then formulate the two-phase parabolic Signorini
problem which has its own applications. We will show how the two phases in this
problem cannot touch. This separation of the phases leads to a proof of the
optimal regularity of the solution.
epshteyn (at) math.utah.edu).
Past lectures: 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.