Applied Mathematics Seminar, Spring 2014

Mondays 3:55 PM - 5:00 PM, LCB 219

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 18214 for Spring 2014. Grading is based on attendance and giving at least one talk presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

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.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148