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January 16 (Friday 3pm, LCB 219) Joint with the Stochastics Seminar.
Speaker: Nicolai Krylov, University of Minnesota
Title: On the regularity properties of conditional densities for partially observable uniformly nondegenerate diffusion processes with Lipshitz coefficients
February 6 (Friday 3:05pm, LCB 215), Joint with Math Bio Seminar.
Speaker: Sarthok Sircar, University of South Carolina
Title: Orientational dynamics of biaxial liquid crystals
Abstract: In 2004, a new "biaxial phase" of liquid crystalline polymer (LCP) was discovered experimentally. Since then, a lot of effort has been devoted at the experimental level to understand the orientational response of such a system in the presence of an external field. To our knowledge, for the first time, we numerically predict and present the various phases of biaxial LCPs and show the sequence of orientations in the different material parameter regions. The talk is divided into two sections. First, we present the steady-state nematics of a simpler class of uniaxial (or spheroidal) LCPs in the presence of an external field, state some theorems regarding the existence of the "solutions" and present the phase bifurcation diagram of the order parameters of these anisotropic systems. The flow-phase sequence of the biaxial (or cuboidal shaped) liquid crystals, in the presence of an external shear flow, are then discussed. The underlying hydrodynamic theory; as well as; the rheological properties are also presented.
Speaker: Yuliya Gorb, Texas A&M University, Mathematics Department
Title: Multiscale Modeling and Simulation of Fluid Flows in Deformable Porous Media
Abstract: The main focus of this talk is on fluid flows in deformable elastic media and associated multiscale problems. Many upscaling methods are developed for flows in rigid porous media or deformable elastic media assuming infinitely small fluid-solid interface displacements relative to the pore size. Much research is needed for the most general and least studied problem of flow in deformable porous media when the fluid-solid interface deforms considerably at the pore level. We introduce a general framework for numerical upscaling of the deformable porous media in the context of a multiscale finite element method. This method allows for large interface displacements and significant changes in pore geometry and volume. For linear elastic solids we present some analysis of the proposed method.
February 11 (Wednesday 3:05pm, LCB 225) Joint with Math Bio Seminar.
Speaker: Margaret Beck, Brown University
Title: Electrical waves in a one-dimensional model of cardiac tissue
Abstract: The electrical dynamics in the heart is modeled by a two-component PDE. Using geometric singular perturbation theory, it is shown that a traveling pulse solution, which corresponds to a single heartbeat, exists. One key aspect of the proof involves tracking the solution near a point on the slow manifold that is not normally hyperbolic. This is achieved by desingularizing the vector field using a blow-up technique. This feature is relevant because it distinguishes cardiac impulses from, for example, nerve impulses. Stability of the pulse is also shown, by computing the zeros of the Evans function. Although the spectrum of one of the fast components is only marginally stable, due to essential spectrum that accumulates at the origin, it is shown that the spectrum of the full pulse consists of an isolated eigenvalue at zero and essential spectrum that is bounded away from the imaginary axis. Thus, this model provides an example in a biological application reminiscent of a previously observed mathematical phenomenon: that connecting an unstable - in this case marginally stable - front and back can produce a stable pulse.
Finally, remarks are made regarding the existence and stability of spatially periodic pulses, corresponding to successive heartbeats, and their relationship with alternans, irregular action potentials that have been linked with arrhythmia.
February 18 (Wed 3:05pm, LCB 215) Joint with Math Bio Seminar
Speaker: Pak-Wing Fok, California Institute of Technology
Title: Acceleration of DNA repair by charge transport: stochastic analysis and deterministic models.
Abstract: A Charge Transport (CT) mechanism has been proposed in several papers (for example see Yavin et al. PNAS 102 3546 (2005)) to explain the colocalization of Base Excision Repair enzymes to lesions on DNA. The CT mechanism relies on redox reactions of iron-sulfur cofactors on the enzyme. Electrons are released by recently adsorbed enzymes and travel along the DNA. The electrons can scatter back to the enzyme to destabilize it and knock it off the strand, or they can be absorbed by nearby lesions and guanine radicals. A stochastic description for the electron dynamics in a discrete model of CT-mediated enzyme kinetics will be presented. By calculating the enzyme adsorption/desorption probabilities, an implicit electron Monte Carlo scheme can be used to simulate the build-up of enzyme density along a DNA strand. Then, a Partial Differential Equation (PDE) model for CT-mediated enzyme binding, desorption and redistribution will be studied. The model incorporates the effect of finite enzyme copy number, enzyme diffusion along DNA and a mean field description of electron dynamics. By computing the flux of enzymes into a lesion, the search time for an enzyme to find a lesion can be estimated. The results show that the CT mechanism can significantly accelerate the search of repair enzymes.
Speaker: Jichun Li, University of Nevada Las Vegas and IPAM, UCLA
Title: Numerical study of Maxwell's equations in negative index metamaterials
Abstract: Since 2000, there has been a growing interest in the study of negative index metamaterials across many disciplinaries. In this talk, I'll first derive the Maxwell's equations resulting from such metamaterials. Then I'll review some time-domain finite element methods recently developed for solving these equations. After that, I'll discuss my most recent work on leap-frog type finite element methods with succinct error estimates. Finally, some numerical results and open issues will be discussed.
March 6 (Friday 1pm, JWB 208, Thesis defense)
Speaker: Lyubima Simeonova, University of Utah
Title: Wave propagation through composite materials: Effective properties and optimization
Abstract: The effective properties of the complex permittivity for waves in one-, two-, and three-dimensional random media are investigated. When the wavelengths of the field are of the same order as the size of the heterogeneities of the composite, scattering effects, such as wave localization and cancellation, must be accounted for. The effective dielectric coefficient is no longer a constant as in the quasistatic case, but a function of the space variable. Since effective dielectric coefficient cannot be calculated explicitly in general, to be useful in applications it is important that we can bound both effective dielectric coefficient itself, and some measure of the spatial variations in effective dielectric coefficient. We have obtained such bounds using novel methods that incorporate probability arguments and the regularity properties of the solutions. The estimates hold in bounded domains for any fixed frequency greater than 0 and show an explicit dependence on the feature size and contrast of the random medium. Pertinent numerical experiments are performed to illustrate the results of the analytical proof. We also consider a related optimization problem of finding the class of materials (described by a probability density function) that minimizes the spatial average of the effective dielectric coefficient. Existence and uniqueness of a minimizing probability density function is proven, and numerical experiments are performed to find the minimizing probability density function. Another optimization problem where there is a restriction in the variability in the medium is solved numerically. The dependence of the effective dielectric coefficient on the contrast in the medium is explored, and series expansion of the effective coefficient, that takes into account the correlation functions of the medium, is derived. Every term in the series is a constant provided the medium is stationary. For media with a correlation function depending on position, the best approximation of the effective dielectric coefficient is a function of the space variable.
The dissertation includes a problem in structural optimization, and in particular optimization of periodic composite structures for sub-wavelength focusing. A slab of material with negative refractive index would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. We pose the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. We show that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition. We use techniques of "topology optimization" in which material distribution is completely arbitrary. We have demonstrated an optimized structure that gives a focus with a spot size 0.284 of the wavelength, which is a significant improvement to those previously obtained by using non-exotic materials.
Speaker: Vianey Villamizar, Brigham Young University, Department of Mathematics
Title: Exact Nonreflecting Boundary Conditions for Multiple Scattering on Generalized Curvilinear Coordinates
Abstract: A multiple scattering problem modelled by the Helmholtz equation is solved. Each arbitrarily shaped scatterer is enclosed by a relatively close artificial boundary. Following [J. Comp. Phys. 201 (2004) 630-650], a DtN boundary condition is derived for several disjoint components of the artificial exterior boundary. Then, a second order finite difference method, combined with the novel Dirichlet-to-Neumann (DtN) non-reflecting boundary condition on generalized curvilinear coordinates, is applied to the inner regions. These inner regions are bounded by the physical scatterer boundaries and the surrounding artificial scatterer boundaries. As a result, the computational cost to obtain a numerical solution is greatly reduced. An approximate solution for multiple scattering from two circular cylinders is obtained using this method. Excellent convergence is obtained for this case when compared to its exact solution. Approximate solutions for more general scatterer configurations of two and three obstacles are also presented. Finally, the radar cross section for various arbitrarily shaped scatterer configurations are obtained.
Speaker: John Willis, DAMTP, University of Cambridge
Title: Effective constitutive relations for waves in composites and metamaterials
Abstract: The description of waves propagating through strongly-heterogeneous material requires some kind of averaging to be performed. Here, the material is taken to be random and ensemble averaging is considered; it is noted in this context that, although the ensemble average is not seen in any one realisation, it nevertheless provides a scaffold upon which the solution in any particular realisation can in principle be built. In practice, resort must be made to approximations. This work establishes exact variational principles which the ensemble averaged solutions must satisfy, and from which "effective constitutive relations" follow. It is demonstrated also that a similar variational structure follows if weighted ensemble averages are employed. Such weighted averages are relevant to so-called metamaterials which contain micro-resonators, whose displacements are best excluded from the averaging process.
March 26 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)
Speaker: Michael Vogelius, Rutgers University
Title: A survey of results concerning existence and blow up for some nonlinear elliptic and parabolic problems related to corrosion modelling
Speaker: Russell Richins, University of Utah (student talk)
Title: Optimal Transportation
Abstract: The problem that inspired the theory of optimal transportation was that of moving a pile of sand to fill a hole when the cost of moving any particle of sand from the pile to the hole is known. The problem is to minimize the cost of filling in the hole. I will discuss the mathematical formulation of this problem as well as some of the tools used to analyze it. I will also show how optimal transportation relates to composite materials in the problem of the optimal placement of conducting material inside an insulating background.
Speaker: Jingyi Zhu, University of Utah
Title: Finite Difference PDE Approaches to Stochastic Volatility Models
Abstract: Stochastic volatility models recognize that the volatility in a stock price by itself is stochastic, and explore various features of the process for the volatility, such as the mean-reverting property. Option prices based on these models are much more realistic when compared to market data, therefore they are widely used by sophisticated option traders. Traditional pricing using stochastic volatility models typically relies on either closed-form solutions or brute force Monte Carlo simulations, with their obvious limitations. Finite difference approaches to solve the resulting time-dependent PDE in two space dimensions provide a powerful alternative, with the advantages such as easy accommodation of variable coefficients and fast numerical convergence. However, one crucial factor that has been illusive is the matter of boundary conditions for the volatility variable. We consider the prototype model for stochastic volatility, the Heston model and its extended forms, supply various boundary conditions accompanied by probabilistic interpretations, and use finite difference techniques to solve the resulting PDE problem in a bounded domain. We present different results, in terms of the market observable "volatility smile" curve, to demonstrate the ramification of the boundary conditions. Comparisons with other approaches such as Monte Carlo simulations are also made to show the advantage of the finite difference approaches.
April 9 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)
Speaker: Gang Bao, Michigan State University
Title: Distinguishability via Uncertainty Principle for inverse Scattering
Abstract: The inverse scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. A general model is concerned with a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain. Given the incident field, the direct problem is to determine the scattered field for the known scatterer. The inverse medium scattering problem is to determine the scatterer from the boundary measurements of near field currents densities. Although this is a classical problem in mathematical physics, numerical solution of the inverse problem remains to be challenging since the problem is nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. In this talk, our recent results in mathematical analysis and computational studies of the inverse boundary value problems for the Maxwell equations will be reported. A novel continuation approach based on the uncertainty principle will be presented. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse medium scattering problems. Convergence issues for the continuation algorithm will be examined. Our most recent progress on inverse source problems will also be discussed.
Speaker: Robert Palais, University of Utah
Title: Rendering with Randomness, Rotating with Reflections
Abstract: Mathematically generated point clouds enhance visualization of surfaces with multiple components along the line of view, provide well behaved grids for numerical methods, and optimize LIDAR analysis via synthesis and simulation. A result from integral geometry suggests an algorithm for representing implicitly defined surfaces. Implementing it involves obtaining uniform random directions on the unit sphere, and we will consider at least seven methods to do so. Converting these directions to uniformly distributed lines in space requires a rotation. We conclude with some surprising algorithms for performing rotations, and their consequences.
April 16 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)
Speaker: Robert Kohn, Courant Institute, New York University
Title: Price Bubbles from Heterogeneous Beliefs
Abstract: Harrison and Kreps showed in 1978 how the heterogeneity of investor beliefs can drive speculation, leading the price of an asset to exceed its intrinsic value. By focusing on an extremely simple market model -- a finite-state Markov chain -- the analysis of Harrison and Kreps achieved great clarity but limited realism. My talk discusses joint work with Xi Chen, which achieves similar clarity with greater realism by considering an asset whose dividend rate is a mean-reverting stochastic process. Our investors agree on the volatility, but have different beliefs about the mean reversion rate. We determine the minimum equilibrium price explicitly; in addition, we characterize it as the unique classical solution of a certain linear differential equation. Our example shows, in a simple and transparent manner, how heterogeneous beliefs about the mean reversion rate can lead to everlasting speculation and a permanent "price bubble".
Speaker: Andrei Kouznetsov, Washington State University, Math Dept.
Title: A Discrete Model of Phase Transitions in Solids
Abstract: We study a discrete model of phase transitions in solids. Our model is a network built of a finite number of nodes connected with non-linear links.
There are many works dedicated to this problem in 1D space. We work in 2D space and this makes the problem significantly more complicated, since we need to satisfy compatibility conditions on the elongations of the links of the network. This compatibility conditions are automatically satisfied in 1D case.
In this presentation the focus is made on the set of deformations of the model with no internal forces. This set is neither a linear space nor a convex space and is very hard to work with. However, this set is the key to understand the properties of materials built on our model.
The current presentation gives a description of the set of deformations with no internal forces, compatibility conditions for elongations, and explains main ideas and proofs of our theoretical results.
Speaker: Peg Howland, Utah State University, Department of Mathematics and Statistics
Title: Using Generalized Discriminant Analysis and Factor Analysis Approximations in Dimension Reduction
Abstract: In applications ranging from text mining to face recognition, dimension reduction is imperative for efficiency. Toward that end, we extend classical linear discriminant analysis (LDA) so that class separability is optimally preserved. The generalized singular value decomposition (GSVD) provides the mathematical framework for this method, and also for a two-stage approach that uses either principal component analysis or QR decomposition before LDA. Further algorithmic simplification can be achieved by applying a rank reduction formula from factor analysis, and restricting the domain to sign or binary vectors. We demonstrate the relationships between these methods, as well as their relative accuracy and complexity, with classification results on document and facial data.