epshteyn (at) math.utah.edu)
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.
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 dierentiable 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.
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.
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.
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.
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.
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.
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.
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.