ANALYSIS/APPLIED MATH SEMINAR
TUESDAY 4:10pm-5:00pm RH 247
Coordinator: Peter Polyakov
Fall 2009
September 15th:
Anton Dzhamay, Department of Mathematics, U of Northern Colorado.
Title: Factorizations
of rational matrix functions with applications to discrete integrable systems
and discrete Painlevé equations
Abstract:We consider the space of
rational matrix functions with a given spectral data (i.e., zeroes and poles of the determinant).
We are interested in ways of representing matrices from this space as products of elementary blocks
and also in determining coordinates on this space that are adapted for such a description.
Interchanging the order of these factors gives an interesting integrable discrete dynamical system,
together with its Lax representation. This approach was pioneered by Moser and Veselov for
polynomial matrices, but it seems that studying rational matrices has certain advantages.
In the quadratic (i.e., two factors corresponding to two simple poles) case we show that this
isospectral system is Lagrangian by writing down an explicit formula for the Lagrangian function.
Furthermore, this approach can be easily adapted to isomonodromic transformations of linear systems
of difference equations. Jimbo and Sakai, and later Arinkin and Borodin, showed that for polynomial
2-by-2 matrices these transformations are described by the difference Painlevé equations.
The use of rational matrices allows us to re-interpret these equations as simple relations between
the eigenvectors of the original and the transformed matrices.
September 29th:
Jeffrey Selden, Department of Mathematics, U of Wyoming.
Title: Some Results
Concerning the Integrated Density of States for Periodic Schroedinger Operators
Abstract:
Periodic Schroedinger operators arise in the quantum mechanical description of solids,
and their spectra are related to physical properties such as whether a given solid acts
as a conductor or an insulator. The integrated density of states function (IDS)
is a sophisticated mechanism for ``counting’’ spectra, and its properties govern
certain large-scale behaviors of the solid. In this talk, I will give a proper formulation
of the IDS and discuss efforts to find its high energy asymptotic expansion: beginning
with the result in one dimension (22 years old), continuing to the results for two
dimensions (4 years old/ 1 year old), and finishing up by describing some recent work
for dimensions greater than two (soon to be born).
October 13th:
Axel Malqvist, Department of Information Technology,
Uppsala University, Sweden.
Title: Multiscale methods for elliptic problems
Abstract: We derive a framework for multiscale approximation of elliptic problems on standard and mixed form. The method is based on a splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. The fine scale approximation is then used to modify the coarse scale equations. A key feature of the method is that symmetry in the bilinear form is preserved in the discrete system. This is different from the variational multiscale method. Other key features are the a posteriori error bounds and the adaptive algorithms for automatic tuning of discretization parameters, based on these bounds. We also present numerical examples where we apply the multiscale method to a problem in oil reservoir simulation.
October 20th:
Man-Chung Yeung, Department of Mathematics, U of Wyoming.
Title: Breakdown analysis of Krylov subspace methods.
Abstract: Krylov subspace methods are popular iterative methods in the real-world computation due to their cheap memory requirement and computational cost. Theoretically, a breakdown of a Krylov method can happen when a zero divisor occurs in its implementation. Practically, however, the phenomenon of breakdown is rarely observed. In this talk, we will try to explain this phenomenon from the probabilistic point of view and show that the probability of breakdown is actually zero.
October 27th:
Long Lee, Department of Mathematics, U of Wyoming.
Title: Numerical investigation of Helmholtz regularizations in a class of partial differential equations.
Abstract: We present a two-step iterative numerical algorithm for studying a class of partial differential equations (PDEs) involving the Helmholtz operator. An equation in this class of PDEs can be written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a midpoint time integrator and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The midpoint time integrator is in fact a symplectic integrator which may be preferable for those PDEs in this class that possess a Hamiltonian structure. To illustrate the efficiency and accuracy of the proposed scheme, we solve several members in the class of PDEs with various initial conditions. Through these examples we compare the regularization effect induced by the Leray-type non-local averaged velocity and that induced by viscosity. We also explore the role of the third-order derivative terms in the regularized Burgers equation. Our numerical investigation on the non-dispersive regularized Burgers equation and two related dispersive PDEs indicates that the proposed method has the ability to handle the dispersive effect of high frequency oscillations efficiently, and to assure the quality of a predicted solution over time. Hence the method is particularly suitable for studying the longtime solution behavior of the proposed class of PDEs, in which many equations are dispersive by nature.
November 3rd:
Hakima Bessaih, Department of Mathematics, U of Wyoming.
Title: On stochastic shell models of turbulence.
Abstract: Different shell models of turbulence will be introduced along with their relationship with the Navier-Stokes equations. Some interesting questions about deterministic models vs stochastic ones will be tackled. Some results on existence and uniqueness of solutions, existence of invariant measures, attracting sets and some other longtime properties.
November 10th:
Pani Fernando, Department of Mathematics, U of Wyoming.
Title: Conformal Martingales.
Abstract: In this talk, we will give a brief introduction about Conformal Martingales. We then discuss some properties of Conformal Martingales and how we can apply Ito formula to Complex valued diffusion processes and Conformal Martingales. Finally, we talk about Conformal Martingale representation in complex setting which is the Complex counterpart of the Martingale representation theorem.
November 17th:
Meng Xu, Department of Mathematics, U of Wyoming.
Title: Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics.
Abstract: In this talk I will formulate a numerical approximation method for the nonlinear filtering of vortex dynamics subject to noise using particle filter method. The convergence of this scheme allowing the observation vector to be unbounded will also be shown.
November 24th:
Farhad Jafari, Department of Mathematics, U of Wyoming.
Title: Strongly Continuous Semigroups and Weighted Composition Operators.
Abstract: Strongly continuous semigroups are fundamental in proving existence of solutions of PDE. After a brief motivation, in this talk we ask and completely answer the following question: Which strongly continuous semigroups arise from weighted composition operators on some spaces analytic functions? Partial results on the analogue of this question in Sobolev spaces will also be presented.
I hope to make this talk accessible to grad students who have had a basic introduction to Complex Analysis, Functional Analysis and PDE.
December 1st:
Chandana Wijeratne, Department of Mathematics, U of Wyoming.
Title: TBA
Last revised November 23, 2009