Modeling and Computation Seminar
Archive

Fall 2007

August

08/23

Organizational Meeting

September

09/06
Alexander O. Korotkevich
Landau Institute for Theoretical Physics, Moscow

Numerical Simulation of Light Propagation in Plasmonic Bragg Gratings

Light propagation in a Bragg periodic structure containing thin films with metallic nanoparticles is studied. Plasmonic resonance frequency, Bragg resonance frequency, and light carrier frequency are assumed to be close. The considered system is nonlinear. Solitary waves solutions were obtained analytically. Numerical simulation allows us to observe light trapping, solitons propagation and interaction.

09/13
Juan M. Restrepo
Department of Mathematics
University of Arizona

A Path Integral Formulation of Data Assimilation

Data assimilation is a forward estimation technique aimed at blending measurement data and a model for the process, yielding what is hoped is an optimal estimate of the solution. Data assimilation schemes tackle realistic scenarios, as they assume that models and measurements may have errors in them; and further, the procedure is not only used to forecast optimal histories, in the case of evolutive problems, it can also be used to pin down or optimize parameter values crucial to the model predictions.

For sequential problems with linear dynamics and Gaussian statistics an optimal assimilator is the well-known Kalman Filter/Smoother. Variants of this technique, extended to nonlinear problems, are used in weather forecasting, hydrology, robotics, and automatic controls.

The path integral method is a sampling-based assimilation scheme that specifically handles nonlinear/non-Gaussian and thus suitable for assimilation in problems where more traditional methods fail. I will describe the technique and demonstrate its performance in the context of financial forecasting and in Lagrangian assimilation in oceanic flows, problems which are shown to lead to estimate failures using more traditional assimilation techniques.

09/20
Robert Indik
Department of Mathematics
University of Arizona

Reconstructing optical phase, products of orthogonal projections and the "angle" between subspaces

In studying an algorithms designed to compute the optical phase based on intensity of an optical field and of the corresponding far field image, I was led to consider the behavior of products of orthogonal projection operations. A careful study of such products is quite interesting and leads to a nice generalization of the notion of angles between lines to that of "angles" between subspaces as well as explaining the local behavior of the algorithm. In this talk I will present the motivating problem (infinite depth of field imaging), and the context in which the linear algebra problem arose, as well as presenting a characterization of operators that are products of two orthogonal projections and how one can define angle between two subspaces.

09/27
Paul Krause
Department of Mathematics
University of Arizona

Data Assimilation through Particle Filters for Small Diffusion Kernels within Branches of Prediction

We consider initial value solutions of a class of Ito nonlinear stochastic ordinary differential equations. A model for small-covariance state fluctuations within branches of a solution is heuristically derived, along with upper bounds for two norms of its covariance matrix for quality control. A particle filter tracing deterministic paths is then proposed for equations with a state-independent diffusion matrix, along with a proper intra-model definition of prediction. The algorithm and its quality control are tested with the noise-perturbed Lorenz equation, showing good results in the additive noise case.

October

10/04
Ildar Gabitov
Department of Mathematics
University of Arizona

Multistability in Optical Waveguide Arrays Containing Negative Index of Refraction Materials

As a result of advances in a small-scale technology, new materials with the unusual property of negative refractive index were recently fabricated. I will discuss the nonlinear properties of such materials. In particular, I will consider the effect of multistability of light propagation in coupled waveguides based on such materials.

10/11
Miroslav Kolesik
College of Optical Sciences
University of Arizona

Dynamics of Driven Interfaces

I will review Monte Carlo methods suitable for simulation of extremely slow processes, such as escape from a metastable state, or motion of a driven interface in the presence of a pinning disorder. The effects of the microscopic dynamics on measurable quantities, for example domain wall mobility, will be discussed as well.

10/18
Christopher Bergevin
Department of Mathematics
University of Arizona

Modeling Cochlear Dynamics

Experiments performed by Georg von Bekesy in the 1930s showed the existence of traveling waves in the inner ear (cochlea). Different spatial locations corresponded to different "best" frequencies, and significant phase accumulation was apparent. Based upon simplifying anatomical assumptions and basic physical laws, we derive a transmission-line model for the cochlea. Due to the variation in impedance along the length, we solve the model using the WKB approximation. This relatively simple approach provides good qualitative agreement with physiological data. However, features that the model fails to capture from viable ears (e.g. nonlinear compression, physiological vulnerability, and otoacoustic emissions) are discussed.

10/25
Paul Dostert
Program in Applied Mathematics
University of Arizona

Uncertainties in the detailed description of reservoir lithofacies, porosity, and permeability are major contributors to uncertainty in reservoir performance forecasting. To make better decisions in reservoir management, this uncertainty must be reduced. We present an approach for sampling permeability (conductivity) conditioned to an integrated response. We consider modifications of a traditional Langevin MCMC method using single-phase upscaling, multiscale methods, and sparse grid collocation. Applications to problems in petroleum engineering and hydrology will be presented.

November

11/01
Tamani Howard
Department of Mathematics
University of Arizona

Sobolev Gradient Preconditioning for a Monge-Ampere Equation

Sobolev gradients are an efficient method of calculating solutions to a wide variety of systems of partial differential equations. Successful applications have been made to problems in transonic flow, Ginsburg-Landau equations for superconductivity, elasticity, minimal surfaces and oil-water separation problems.

I will discuss why the poor numerical performance of ordinary gradients and the good performance of Sobolev gradients give an instance of the first law of numerical analysis: "Analytical Difficulties and Numerical Difficulties Always Come in Pairs."

I will show how the method can also be successfully applied to the hyperbolic fully non-linear Monge-Ampere equation:

Det(D^2 z) = \kappa (1 + {z_x}^2 + {z_y}^2)^2

on the unit square, where \kappa = -1, represents the Gaussian curvature of the surface described by z.

11/08
Misha Stepanov

On a Failed Attempt to Construct an [183, ?, 50] Error-Correcting Code

The iterative decoding of a certain class of error correcting codes, namely LDPC codes, works best if the underlying graph structure locally looks like a tree. The bigger the tree-like vicinity of the vertex in the graph, the better the guaranteed performance of the iterative decoding. I was trying to come up with an error correcting code of 183 bits and a certain local tree structure. This was proved to be impossible because of a certain fact from finite projective geometry.

11/15
Kevin Lin

Markov Chain Monte Carlo, Couplings, and Variance Reduction

In many problems involving Monte Carlo-type simulations, the target distribution is specified only implicitly as the invariant distribution of a Markov chain. Examples include stochastic models of nonequilibrium transport processes, queues, and chemical reactions. The lack of explicit expressions for the target distribution makes it difficult to apply standard variance reduction methods that rely on changing the dynamics, e.g. multigrid Monte Carlo. In this talk, I will describe a class of methods which can improve the accuracy of such calculations in certain situations, and illustrate the method on a simple stochastic lattice gas model, the symmetric simple exclusion process. The method is based on coupling the Markov process of interest to a second, closely-correlated process with known invariant distribution. This is joint work with Jonathan Goodman.

11/29
Brad Story
Department of Speech, Language, and Hearing Sciences
The University of Arizona

Simulation of Speech and Singing Based on a Model of the Time-Varying Vocal Tract Shape

During speech or singing production, the human vocal tract is a nearly continuously changing conduit through which sound propagates. As the vocal tract shape changes, its resonance frequencies also continuously vary, shaping an excitation signal into a sequence of vowels and consonants. This presentation will focus on modeling the time-dependent movements of the vocal tract shape and their acoustic consequences. Based on data collected with MRI and x-ray microbeam techniques, a kinematic model of the vocal tract area function has been developed that allows for efficient specification of time-dependent cross-sectional area changes in an acoustic waveguide. When coupled with a voice source, the result is a basic simulation of the sound production process from which pressures and airflows are generated. The components of the model will be presented and then used to demonstrate some time-dependent relations between the vocal tract shape and resulting acoustic characteristics.