| Professor Dinh-Liem Nguyen Department of Mathematics, Kansas State University |
| Title: Direct and Inverse Electromagnetic Scattering Problems for Bi-Anisotropic Media |
| Abstract. We present in this talk our study on direct and inverse scattering of time-harmonic electromagnetic waves from bi-anisotropic media. For the direct problem, we study an integro-differential equation formulation, its Fredholm property, and uniqueness of weak solution. Using this integro-differential formulation we present a fast spectral Galerkin method for the numerical solution to the direct problem. We solve the inverse problem of recovering bi-anisotropic scatterers from far field data using orthogonality sampling methods. These methods aim to construct imaging functionals which are robust with noise, computationally cheap, and require data for only one or a few incident fields. |
Friday, May 10, 11:00AM-12:00 Noon, Conference room
Categories: Updates
Monday, April 1, 11:00AM-12:00 Noon, Fretwell 315
Categories: Updates
| Professor Khai Nguyen Dept. of Mathematics, NC State University |
| Title: Kolmogorov Entropy Compactness Estimates for nonlinear PDEs |
| Abstract. Inspired by a question posed by Lax in 2002, in recent years it has received an increasing attention the study on the quantitative analysis of compactness for nonlinear PDEs. In this talk, I will present recent results on the sharp compactness estimates in terms of Kolmogorov epsilon entropy for hyperbolic conservation laws and Hamilton-Jacobi equations. Estimates of this type play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of “resolution” of a numerical method for the corresponding equations. |
Wednesday, March 27, 3:00PM-4:00PM, Conference room
Categories: Updates
| Professor Ernst Presman, Central Economics Mathematics Institute, Russian Acad. of Sci., Moscow, Russia |
| Title: Markov Chain Modulated Inventory Model |
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Abstract: In her PhD thesis [1], Jennifer Hill, a graduate of UNCC, analyzed the following model proposed by I. Sonin (see also [2] and [3]). There is a firm, which uses a certain commodity for production and consumes it with a unit intensity. The price of the commodity follows a continuous time Markov chain with a finite number N of states and known transition rates. The firm can keep some of the commodity in storage. At any time point, the firm can either purchase the commodity at the current price or use some of its stored reserves. Further, it can buy the commodity either with some intensity or instantly some amount for storage. The storage cost is proportional to the amount of the commodity stored. The goal is to minimize the average (or discounted) performance cost, which equals the storage cost plus the purchase cost. For N = 2 and for some cases with N = 3, Hill and Sonin found the minimal values of thresholds in the class of threshold strategies. We consider the general case and prove that the optimal strategy is indeed the threshold one. Further, we give an algorithm for sequential construction of optimal thresholds beginning from the smallest one. References [1] J. Hill, (2004), A Markov-Modulated Acquisition Strategy, PhD thesis. [2] J. Hill, I. Sonin, (2006). An Inventory Optimization Model with Markov Modulated Commodity Prices, abstract, Intern. Conf. on Management Sciences, Univ.of Texas at Dallas. [3] M. Katehakis, I. Sonin, (2013), A Markov Chain Modulated Inventory Model, abstract, INFORMS, 2013. |
Friday, April 12, 11:00AM-12:00 Noon, Fretwell 122
Categories: Updates
| Professor Rakesh Dept. of Mathematical Sciences, University of Delaware |
| Title: Inverse problems for hyperbolic PDEs |
| Abstract: We state and motivate (oil prospecting, medical imaging, radar, etc.) some inverse problems for the wave equation. We then describe some of the important theoretical results and end with two important ideas in proving these results. |
Friday, March 29, 2:00PM-3:00PM, Fretwell 118
Categories: Updates
| Professor Julie Mitchell,Bioscience Division, Oak Ridge National Lab |
| Title: Feature Selection in Biomolecular Models |
| Abstract: Protein-protein interactions regulate many essential biological processes and play an important role in health and disease. The process of experimentally characterizing protein residues that contribute the most to protein-protein interaction affinity and specificity is laborious. Thus, developing models that accurately characterize hotspots at protein-protein interfaces provides important information about how to drug therapeutically relevant protein-protein interactions. In this work, we combined the KFC2a protein-protein interaction hotspot prediction features with Rosetta scoring function terms and interface filter metrics. A 2-way and 3-way forward selection strategy was employed to train support vector machine classifiers, as was a reverse feature elimination strategy. From these results, we identified subsets of KFC2a and Rosetta combined features that show improved performance over KFC2a features alone. The forward selection algorithm also helped elucidate the biophysical principles that determine whether a given amino acid is a binding hot spot. |
Monday, Feb 11, 11:00AM-12:00 Noon, Conference room
Categories: Updates
| Professor Hyungchun Lee of Dept. of Mathematics, Ajou University, South Korea |
| Title: Uncertainty quantification for partial differential equations and their optimal control problems |
| Abstract: In this talk, we consider UQ (Uncertainty quantification) and optimal control problems for partial differential equation with random inputs. First we introduce a general approach of studying UQ and then consider some optimal control problems.
To determine an applicable deterministic control $\hat{f}(x)$, we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility. |
Wednesday, Jan 16, 11:00AM-12:00 Noon, Conference room
Categories: Updates
| Inbo Sim, University of Ulsan, South Korea |
| Title: Symmetry-breaking bifurcation for the one-dimensional H\'{e}non and Moore-Nehari differential equations |
| Abstract: We show the existence of a symmetry-breaking bifurcation point for the one-dimensional H\'{e}non and the Moore-Nehari differential equation.
Employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Moreover, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) and show that a bifurcation point for Moore-Nehari equation is explicitly represented as a function of \p\ which is an exponent of nonlinear term. |
Monday, Nov 26, 11:00AM-12:00 Noon, Fretwell 315
Categories: Updates
| Grace Stadnyk, NC State University |
| Title: Some Combinatorial Results on the Edge-Product Space of Phylogenetic Trees |
| Abstract: I will discuss a topological space that arises in evolutionary biology called the edge-product space of phylogenetic trees. In particular, I will discuss some of the combinatorial properties of a particularly natural CW decomposition of this space. It is known that intervals in the resulting face poset are shellable, but the manner in which the maximal faces intersect is still not well-understood. In particular, the poset in its entirety is not known to be shellable. I will introduce a partial order on the maximal faces of the edge-product space of phylogenetic trees called the enriched Tamari poset, which can be viewed as a generalization of the Tamari lattice. I will use this poset to show that the edge-product space of phylogenetic trees is gallery-connected. I will conclude by discussing the question of whether the face poset of this topological space is shellable. I will define most of the notions that appear in the talk, so it should be accessible to all. |
Monday, Nov 12, 11:30AM-12:30 PM, Fretwell 315
Categories: Updates
| Dr. Min Hyung Cho, University of Massachusetts at Lowell |
| Title: Fast Integral equation methods for wave scattering in layered media |
| Abstract: Many modern electronic/optical devices rely on waves such as solar cells, antennae, radar, and lasers. These devices are mostly built on a patterned layered structure. For optimizing and characterizing these devices, numerical simulations play a crucial role. In this talk, an integral equation method in 2- and 3-D layered media Helmholtz equation will be presented. In 2-D, the boundary integral equation with the periodizing scheme is used. This method uses near- and far-field decomposition to avoid using the quasi-periodic Green’s function. By construction, the far-field contribution can be compressed using Schur complement with minimal computational cost. The new method solved the scattering from a 1000-layer with 300,000 unknown to 9-digit accuracy in 2.5 minutes on a workstation. In 3-D, a Lippmann-Schwinger type volume integral equation is used with layered media Green’s function to include interface condition between layers and reduces the problem to only scatterers.
In both 2- and 3-D layered media, a fast integral operator application is required because integral equation methods usually yield a dense matrix system. A heterogenous fast multipole method (H-FMM) is developed. This is a hierarchical method and uses recursively-generated tree-structure. The interactions from far fields are compressed with free-space multipole expansion. All the spatially variant information are collected into the multipole-to-local translation operators. As a result, many free-space tools can be adapted directly without any modification to obtain an optimal O(N) algorithm for low frequency.
This is a joint work with Jingfang Huang (UNC), Alex Barnett (Dartmouth College), Duan Chen (UNC Charlotte), and Wei Cai (Southern Methodist University)
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Friday, Nov 2, 11:30AM-12:30 PM, Fretwell 116
Categories: Updates
| Dr. Dat Cao, Texas Tech University |
| Title: Asymptotic expansions for solutions of Navier-Stokes equations in a general class of decaying forces |
| Abstract: We discuss the long time behavior of solutions to the three-dimensional Navier-Stokes equations with periodic boundary conditions. We introduce appropriate systems of decaying functions and the asymptotic expansion with respect to those systems. It is shown that if the force has a long-time asymptotic expansion in Sobolev-Gevrey spaces in such a general system then any Leray-Hopf weak solution admits an asymptotic expansion of the same type. Particularly, we obtain the expansions in terms of the logarithmic and iterated logarithmic decay and recover the case of power decay obtained earlier. This is a joint work with Luan Hoang |