Seminars and Colloquiums
for the week of April 24, 2017
Greg Bell, University of North Carolina at Greensboro, Monday
Erkan Nane (Auburn University), Tuesday
Ajay Jasra (National University of Singapore), Tuesday
Professor Chris Rodger, Auburn, Tuesday
Andrew Marchese, UTK, Wednesday
Professor Reza Abedi, UT Space Institue, Tullahoma, TN, Wednesday
Kylie Berry, UTK, Thursday
Joanna Furno, Indiana University-Purdue University Indianapolis (IUPUI), Friday
Sue Brenner, LSU, Friday
TEA TIME -
3:00 pm – 3:30 pm
Monday, Tuesday, and Wednesday: Ayres 401
Hosted by: Maggie, Kylie and Brittany
Monday, April 24th
Title: On Product Stability of Asymptotic Property C
Speaker: Greg Bell, University of North Carolina at Greensboro
Time: 2:30p - 3:20p
Room: Ayres 113
Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver's topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces. We also show that metric free products preserve asymptotic property C. Of particular interest is the case of the direct product and free product of countable groups in proper metrics. Our results settle a question Dydak and Virk, and a question from the Lviv Topology Seminar. This is joint work with Andrzej Nagórko (University of Warsaw).
Tuesday, April 25th
STOCHASTICS/ PROBABILITY SEMINAR
TITLE: Analysis of space-time fractional stochastic partial differential equations
SPEAKER: Erkan Nane (Auburn University)
ROOM: 113 Ayres
Stochastic partial differential equations and random fields have been used as successful models in various areas of applied mathematics, statistical mechanics, theoretical physics, theoretical neuroscience, theory of complex chemical reactions, fluid dynamics, hydrology, cosmology, mathematical finance, and other scientific areas. In this talk I will consider nonlinear space-time fractional stochastic heat type equations. These time fractional stochastic heat type equations are attractive models that might be used to model phenomenon with random effects with thermal memory. In this talk, I will discuss: (i) Existence and uniqueness of solutions and existence of a continuous version of the solution; (ii) absolute moments of the solutions of this equation grows exponentially; and (iii) intermittency fronts. Our results are significant extensions of those in recent papers by Foodun, Liu, Omaba (Moment bounds for a class of fractional stochastic heat equations. Preprint. 2014), Foondun and Khoshnevisan(Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568), and Conus and Khoshnevisan (On the existence and position of the farthest peaks of a family of stochastic heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681--701)
TITLE: Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo
SPEAKER: Ajay Jasra (National University of Singapore)
ROOM: 113 Ayres
In this talk consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler Maruyama method. Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC to implement a new approximation of the multilevel (ML) Monte Carlo (MC) collapsing sum identity. Our approach comprises constructing an approximate coupling of the posterior density of the joint distribution over parameter and hidden variables at two different discretization levels and then correcting by an importance sampling method. The variance of the weights are independent of the length of the observed data set. The utility of such a method is that, for a prescribed level of mean square error, the cost of this MLMC method is probably less than i.i.d.~sampling from the posterior associated to the most precise discretization. However, the method here comprises using only known and efficient simulation methodologies. The theoretical results are illustrated on numerical examples.
Title: Amalgamations and Hamilton Decompositions
Speaker: Professor Chris Rodger, Auburn
Time: 3:40p – 4:30p
Room: Ayres 401
In this talk, we will explore the use of amalgamations in the construction of graph decompositions, most often looking for hamilton cycle decompositions. This method uses graph homomorphisms to envision an "outline" of the structure of interest, then attempts to disentangle the merging of new vertices created by the homomorphism in such an outline structure. As will be shown, this method has proved to be very effective, for example, in the studying the embedding of edge-colorings of graphs into hamilton decompositions. Fair division of colors in factorizations of graphs is also a very useful property that crops up in this setting, surprisingly even leading to symmetric versions of Sudoku squares.
Wednesday, April 26th
PhD THESIS DEFENSE
TITLE: Data Analysis Methods using Persistence Diagrams
SPEAKER: Andrew Marchese, UTK
ROOM: Ayres 111
In recent years, persistent homology techniques have been used to study data and dynamical systems. Using these techniques, information about the shape and geometry of the data and systems leads to important information regarding the periodicity, bistability, and chaos of the underlying systems. In this thesis, we study all aspects of the application of persistent homology to data analysis. In particular, we introduce a new distance on the space of persistence diagrams, and show that it is useful in detecting changes in geometry and topology, which is essential for the supervised learning problem. Moreover, we introduce a clustering framework directly on the space of persistence diagrams, leveraging the notion of Frechet means. Finally, we engage persistent homology with stochastic filtering techniques. In doing so, we prove that there is a notion of stability between the topologies of the optimal particle filtered path and the expected particle filtered path, which demonstrates that this approach is well posed. In addition to these theoretical contributions, we provide benchmarks and simulations of the proposed techniques, demonstrating their usefulness to the field of data analysis.
Committee members are: Vasileios Maroulas (Chair), Haielab Hilafu, Michael Langston, Jan Rosinski
COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR
Title: An asynchronous spacetime discontinuous Galerkin method for the solutions of hyperbolic PDEs
Speaker: Professor Reza Abedi, UT Space Institue, Tullahoma, TN
Time: 3:35p - 4:35p
Room: Ayres 113
The multi-scale, multi-physics, and complex nature of many modern problems as well as rapid advances in supercomputing call for more accurate, stable, and scalable computational methods. First, I will briefly discuss and compare a few popular numerical methods such as continuous and discontinuous Galerkin (DG) finite element methods in terms of their accuracy, efficiency, and stability for discontinuous solution features and dynamic problems. Some metrics for this discussion will be on high order methods, adaptive schemes, and parallel computing.
Next, I will present the Spacetime Discontinuous Galerkin (SDG) finite element method, in which space and time are directly discretized by elements that employ discontinuous basis functions. The SDG’s element-wise balance property, local solution scheme, and inherent stability yield a method that strongly satisfies the aforementioned criteria and delivers exceptionally accurate results for problems with severe nonlinearities and singular features. Examples will be shown from elastodynamics, electromagnetics, and thermal and fluid mechanics. For the electromagnetics problem, the continuum and discrete expressions of Maxwell’s equations for a linear dispersive media are presented. Numerical solutions are used to verify the method’s energy stability condition and convergence rate. In addition, the effect of the choice of target fluxes, which are an important aspect of DG methods, is discussed. Finally, the adaptive SDG method is combined with a parameter retrieval scheme to characterize dispersive response of certain composites.
Thursday, April 27th
DIFFERENTIAL EQUATIONS SEMINAR
Title: 1-D Energy-Based Blending of Peridynamics and Classical Elasticity
Speaker: Kylie Berry, UTK
Time: 2:00p – 3:00p
Room: Ayres 113
Classical Elasticity and Peridynamics are different mathematical frameworks used in the study of continuum mechanics. Classical Elasticity is a computationally cost effective method of modeling, but because of it use of PDEs it assumes some degree of smoothness to the deformation. On the other hand, Peridynamics, through its use of integrals rather than PDEs, can deal with both continuous and discontinuous material deformations without distinction, but it is more computationally expensive. Since both methods have “opposite” strengths and weaknesses, many people are working on coupling the two methods effectively. I am currently working on an energy-based blending of these two methods. While I am not the first to consider this type of blending, it is my goal to show, in a mathematically rigorous way, the validity of this model. In this talk we will discuss both the work that I have done and the work I plan to do to these ends.
Friday, April 28th
TOPOLOGY/ GEOMETRY SEMINAR
Title: Measure and Hausdorff dimension for p-adic Julia sets
Speaker: Joanna Furno, Indiana University-Purdue University Indianapolis (IUPUI)
Time: 1:25p - 3:15p
Room: Ayres 114
Haar measure and Hausdorff dimension are two possible methods of measuring size in the field of p-adic numbers and its finite extensions. We first explore the Haar measure and Hausdorff dimension for balls in a finite extension of the p-adic numbers. Then we use these two tools to measure the size of the Julia set for some p-adic repellers. Finally, we give some concrete polynomial examples from among these p-adic repellers.
Title: Finite Element Methods for Fourth Order Elliptic Variational Inequalities
Speaker: Sue Brenner, LSU
Time: 3:30pm – 4:30pm
Room: Ayres 405
Fourth order elliptic variational inequalities appear in obstacle problems for Kirchhoff plates and optimization problems constrained by second order elliptic partial differential equations. The numerical analysis of these variational inequalities is more challenging than the analysis in the second order case because the complementarity forms of fourth order variational inequalities only
exist in a weak sense. In this talk we will present a unified framework for the a priori analysis of finite element methods for fourth order elliptic variational inequalities that are applicable to C1 finite element methods, classical nonconforming finite element methods, and discontinuous Galerkin methods.
If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.
If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu
3/13/17 - Spring Break