Department of Mathematics

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Past talks for Fall 2012

August 24 (Friday 4-5pm, LCB 222, student talk)
Speaker: Andy Thaler, University of Utah, Mathematics Dept.,
Title: Bounds on the Average Fields and Volume Fraction in Two-Phase Composite with Complex Permittivities
Abstract: Raĭtum (1983) and Tartar (1995) derived bounds on the average displacement current field in a composite with real permittivity. Geometrically, these bounds are represented as a disk (ball) in two (three) dimensions. The present work extends these bounds to the case of complex-valued dielectric constants in composites with two isotropic phases; the bounds we derive correlate the average electric and displacement field values and the volume fractions of the phases. We utilize an extension of the splitting method introduced by Milton and Nguyen (2012) rather than variational principles. These bounds may have important design applications related to directing fields. The above bounds can also be used in an inverse fashion to bound the volume fraction of one of the phases from measurements of the average electric field and average displacement field. Our bounds generalize to the case of a single inclusion in a body - in particular, boundary measurements of the complex potential and flux can be used to estimate the volume fraction of the inclusion. These bounds could have applications in non-destructive testing and medicine, such as in the screening of organs prior to transplantation.

August 27
Speaker: Yongtao Zhang, University of Notre Dame, Dept. of Applied and Computational Mathematics and Statistics
Title: Computational methods in pattern formation solutions
Abstract: In this talk, I will present two kinds of numerical methods for mathematical models in biological pattern formation problems. The first method is the weighted essentially non-oscillatory (WENO) method for solving the nonlinear chemotaxis models. Chemotaxis is the phenomenon in which cells or organisms direct their movements according to certain gradients of chemicals in their environment. Chemotaxis plays an important role in many biological processes, such as bacterial aggregation, early vascular network formation, among others. While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this work, we combined two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. The second method is the Krylov implicit integration factor (IIF) method for nonlinear reaction-diffusion and advection-reaction-diffusion equations in pattern formations. Integration factor methods are a class of "exactly linear part" time discretization methods. Efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. Here, I will present our results in solving this problem by applying the Krylov subspace approximations to the matrix exponential operator. We applied this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction-diffusion equations. Then we extended the Krylov IIF method to solve advection-reaction-diffusion PDEs and achieved high order accuracy. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the methods.

October 15
Speaker: Andrej Cherkaev, University of Utah, Mathematics dept.
Title: Lattice composites and damage
Abstract: The talk discusses mesoscale behavior of inhomogeneous and damageable lattice structures, compatibility conditions, damage spread, "lattice composites", and a range of their effective properties.

October 22
Speaker: Gregory J. Rodin, The University of Texas at Austin, Institute for Computational Engineering and Sciences.
Title: Effective properties of discrete and continuum systems
Abstract: In multiscale modeling, the transition from fine to coarse scales invariably gives rise to effective (or macroscopic) properties and governing equations. This talk will address two topics. In the first part of the talk, we consider coarse-graining of n-dimensional first order linear systems, and demonstrate how the dynamics of eliminated degrees of freedom (slaves) affects that of the masters. In particular, we establish that the dynamics of coarse-grained system is governed by qualitatively different properties and equations. In the second part of the talk, we discuss a thermodynamic limit associated with the determination of the effective properties of conducting composites formed by particles embedded in a matrix. We establish that, under the assumption that the particles do not interact, the effective properties are consistent with Maxwell-Clausius-Mossotti formula, which is closely related to the variational bounds.

October 29
Speaker: Daniel Onofrei, University of Houston
Title: Active control of acoustic and electromagnetic fields
Abstract: The problem of controlling acoustic or electromagnetic fields is at the core of many important applications such as, energy focusing, shielding and cloaking or the design of super-directive antennas. The current state of the art in this field suggests the existence of two main approaches for such problems: passive controls, where one uses suitable material designs to control the fields (e.g., material coatings of certain regions of interest), and active control techniques, where one employs active sources (antennas) to manipulate the fields in regions of interest. In this talk I will first briefly describe the main mathematical question and its applications and then focus on the active control technique for the scalar Helmholtz equation in a homogeneous environment. The problem can be understood from two points of view, as a control question or as an inverse source problem (ISP). This type of ISP questions are severely ill posed and I will describe our results about the existence of a unique minimal energy solution. Stability of the solution and extensions of the results to the case of nonhomogeneous environment and to the Maxwell system are part of current work and will be described accordingly.

October 31 (Oral exam, Wed 4-5pm in JTB 110)
Speaker: Michał Kordy, University of Utah
Title: Compatible finite element formulation of the equation for electric field in the frequency domain: crucial role of the divergence correction.
Abstract: A novel method for the 3-D diffusive electromagnetic (EM) forward problem is developed and tested. The method considers the standard curl curl equation for the electric field E, a finite element formulation, and the divergence correction. For E field, edge elements are considered which are compatible with the curl operator. Elements are tetrahedra with straight edges, which allow for a natural definition of scalar fields space, used in the divergence correction -- the space of piecewise linear functions. Together with edge elements, these spaces are part of a finite de Rham diagram. With the aid of properties of this diagram, namely Hodge decomposition on a finite grid level, the equation for E is decoupled into two equations: one on the range of the gradient, and the other on a space orthogonal to it. The eigenvalues associated with the equation of the range of the gradient are much smaller than eigenvalues associated with the space orthogonal to it. As a result the system matrix is ill-conditioned and Krylov solvers will focus on imposing an equation on the space orthogonal to the gradient, yet will struggle with imposing the equation on the range of the gradient. This highlights the importance of the divergence correction which readdresses this tendency. Numerical study shows that divergence correction on one hand acts as a preconditioner, speeding up the convergence, and on the other hand corrects the obtained solution. This correction is very important for low frequencies especially in the air, where if the correction is not applied properly, the field differs greatly from the true solution even if the residual of the equation appears small.
The talk is a part of an oral qualifying exam for the PhD student.

November 5
Speaker: Bei Wang, University of Utah, SCI Institute
Title: Stratification Learning through Local Homology Transfer
Abstract: Advances in scientific and computational experiments have increased our ability to gather large collections of data points in high-dimensional spaces, far outpacing our capacity to analyze and understand them. For instance, in a large-scale simulation, one might want to understand the relationships between a large number of input parameters and their effects on a set of particular outcomes.
We approach the problem as follows, given a point cloud of data sampled from some underlying space, can we infer the topological structure of the space? Often we assume the support of the domain is either from a low- dimensional space with manifold structure, or more interestingly, contains mixed dimensionality and complexity. The former is a classic setting in manifold learning. The latter can often be described by a stratifi ed set of manifolds and becomes a problem of particular interest in the fi eld of strati fication learning.
Stratified spaces, while not manifolds, can be decomposed into manifold pieces that are glued together in some uniform way. An important tool in strati cation learning is the study of local spaces, that is, the neighborhoods surrounding singularities, where manifolds of different dimensionality and complexity intersect.
We show that point cloud data can under certain circumstances be clustered by strata in a plausible way. For our purposes, we consider a stratified space to be a collection of manifolds of different dimensions which are glued together in a locally trivial manner inside some Euclidean space. To adapt this abstract definition to the world of noise, we first define a multi-scale notion of stratified spaces, providing a stratification at different scales which are indexed by a radius parameter. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata based on local homology transfer maps. We prove a correctness guarantee for this clustering method under certain topological conditions. We then provide a probabilistic guarantee for the clustering for the point sample setting - we provide bounds on the minimum number of sample points required to state with high probability which points belong to the same strata. Finally, we give an explicit algorithm for the clustering.
If time permits, I would mention some other work in high-dimensional data analysis and visualization.

November 12 (exceptionally in LCB 219)
Speaker: Michael Mascagni, Florida State University, Department of Computer Science
Title: Novel Stochastic Methods in Biochemical Electrostatics
Abstract: Electrostatic forces and the electrostatic properties of molecules in solution are among the most important issues in understanding the structure and function of large biomolecules. The use of implicit-solvent models, such as the Poisson-Boltzmann equation (PBE), have been used with great success as a way of computationally deriving electrostatics properties such molecules. We discuss how to solve an elliptic system of partial differential equations (PDEs) involving the Poisson and the PBEs using path-integral based probabilistic, Feynman-Kac, representations. This leads to a Monte Carlo method for the solution of this system which is specified with a stochastic process, and a score function. We use several techniques to simplify the Monte Carlo method and the stochastic process used in the simulation, such as the walk-on-spheres (WOS) algorithm, and an auxiliary sphere technique to handle internal boundary conditions. We then specify some optimizations using the error (bias) and variance to balance the CPU time. We show that our approach is as accurate as widely used deterministic codes, but has many desirable properties that these methods do not. In addition, the currently optimized codes consume comparable CPU times to the widely used deterministic codes. Thus, we have an very clear example where a Monte Carlo calculation of a low-dimensional PDE is as fast or faster than deterministic techniques at similar accuracy levels.

November 26
Speaker: Qinghai Zhang, University of Utah, Mathematics Dept.
Title: A Cocktail of ODE, Differential Geometry, Jordan Curve Theorem, Fluid Dynamics, Finite Volume Methods, Semi-Lagrangian Methods and More
Abstract: Given a time interval (tn, tn+k) and a fixed simple curve LN in the time-dependent velocity field, which set of particles will pass through LN and contribute to its flux within the time interval? The locus of these fluxing particles at time tn is equivalent to the donating region (DR), the dependence domain of the fixed curve for the advection equation. In this talk, I propose an explicit, constructive, and analytical definition of DR based on classical ODE theory, differential geometry, Jordan curve theorem, and several well-known characteristic curves in fluid dynamics. I will show that the DR contain, and only contain, all the particles that has a net effect of passing through LN once.
The second half of this talk will focus on one computational aspect of DR, namely the algorithms of Lagrangian flux calculation (LFC) via algebraic quadratures over the spline-approximated DRs. As DR connects to both Eulerian and Lagrangian viewpoints, LFC naturally leads to a conservative semi-Lagrangian method whose time step size is free of the Eulerian stability constraint of Courant number being less than one. Various numerical tests on 2D advection tests demonstrate high-order accuracies from the 2nd order to the 8th order (both in time and in space) and large Courant numbers from 1 to 10000.

December 3
Speaker: Semyon Tsynkov, North Carolina State University, Department of Mathematics
Title: Dual carrier probing for spaceborne SAR imaging
Abstract: Imaging of the Earth's surface by spaceborne synthetic aperture radars (SAR) may be adversely affected by the ionosphere, as the temporal dispersion of radio waves gives rise to distortions of signals emitted and received by the radar antenna. Those distortions lead to a mismatch between the actual received signal and its assumed form used in the signal processing algorithm (known as the matched filter). In turn, the discrepancy between the filter and the signal causes deterioration of the image.
To mitigate the ionospheric distortions, we propose to probe the terrain, and hence the ionosphere, on two distinct carrier frequencies. The resulting two images appear shifted with respect to one another, and the magnitude of the shift allows one to evaluate the total electron content (TEC) in the ionosphere. Knowing the TEC, one can correct the matched filter, and hence improve the quality of the image. Robustness of the proposed approach can subsequently be improved by applying an area-based image registration technique to the two images obtained on two frequencies. The latter enables a very accurate evaluation of the shift, which, in turn, translates into a very accurate estimate of the TEC.
Time permitting, we will also discuss how the Ohm conductivity of the ionosphere may affect the SAR resolution, what may be the effect of the ionospheric turbulence, how to take into account the horizontal variation of the ionospheric parameters, and what may be the role of the anisotropy due to the magnetic field of the Earth.
Joint work with E. Smith and M. Gilman.

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