Mathematical Sciences

School of Natural Sciences & Mathematics

Computational Science Seminar F14

Dec 5 Andrea Barreiro Mathematics, SMU Dynamics and complexity of neural spike correlations Correlations among neural spike times are found widely in the brain; they can be used to modulate or limit information in population coding, and open the possibility for cooperative coding of sensory inputs across neural populations. Correlations also introduce a daunting complexity; when every neuron is potentially correlated with every other, the amount of information needed to represent spiking activity grows exponentially with the number of cells.

In this talk I discuss recent work towards understanding how the structure and transfer of correlated activity is affected by both intrinsic neuron dynamics and network architecture. I first present an interesting and non-intuitive result about how the phase space structure of neural models – specifically the bifurcation that mediates their transition from rest to firing – affects their ability to transmit common signals. Second, I analyze the ability of pairwise maximum entropy models – a technique borrowed from statistical mechanics for representing spiking activity in a simpler
way – to perform on a broad class of feedforward circuits. This study provides an explanation for the surprising finding that responses in primate retinal ganglion cells are well-described by this model, even in cases where the circuit architecture seems likely to create a richer set of outputs (Shlens et al., J Neurosci, 2006; 2009; Schneidman et al., Nature, 2006), and identifies pathways by which specific circuit mechanisms influence the complexity of correlation structure.
Nov 14 Adrianna Gillman CAAM, Rice Fast direct solution techniques for elliptic partial differential equations
In many areas of science and engineering, the cost of solving a linear boundary value problem determines what can and cannot be modeled computationally. In some cases, it is possible to recast the problem as an integral equation which sometimes leads to a reduction in the dimensionality of the problem. Alternatively, the differential equation can be discretized directly with a finite element or finite difference method. Either way, one is left with having to solve a large linear system. The computational cost of directly inverting the linear system via Gaussian elimination grows as
O(N^3) where N is the size of the system. Due to recent developments (multigrid, FMM, FFT, etc.), there are fast methods for most of these linear systems of equations. By fast, we mean that the computational cost of solving the problem grows as O(N log^k N) where k is a small integer, normally k = 0, 1, or 2. Many fast schemes are based on iterative techniques that build a sequence of approximate solutions that converges to the exact solution and often require the use of a problem specific preconditioner. In this talk, we will present methods that directly invert the system by exploiting structure in the matrix with a cost that grows linearly with the problem size. Such fast direct methods are more robust, versatile and stable than iterative schemes. They are also much faster for problems with multiple right-hand sides.
Nov 7 Russell Hewett Total E&P Research and Technology USA A polarized-trace solver for the Helmholtz equation
Full-waveform inversion is a method for recovering Earth’s physical parameters by matching seismic observations with geophysical simulations. To avoid issues due to nonconvexity in full-waveform inversion, the problem is treated in the frequency domain. Frequency domain imaging requires scalable solvers for the Helmholtz equation to make feasible imaging in high resolution and in 3D, however, this remains an open problem. We present recent developments on a domain-decomposed preconditioner for the acoustic Helmholtz equation, based on the notion of polarization, that demonstrates substantial scalability and performance improvements over the state-of-the-art. I will also introduce PySIT, an open source seismic inversion toolbox written in Python, which is designed for rapid prototyping and reproducible research.
Oct 31 William Frensley Electrical Engineering, UTD The Many Levels of Semiconductor Device Physics
I will present a very quick tour of the logical chain that is presumed to exist between fundamental physics and active device technology, showing how the mathematical structures change with physical length scale, and illustrating the behavior at each level with simple numerical solutions and simulations. The macroscopic description consists of a set of partial differential equations. The point to be made is that the conventional procedure (taking those equations to define a problem in modern analysis and constructing numerical solutions as dictated by asymptotic error analysis) produces a confidence in the results which is entirely unwarranted. Discrete numerical models, constructed by the simplest possible procedures directly from the physical system, come much closer to the true behavior of the device, because the discretization “errors” tend to correct the deficiencies in the macroscopic equations.
Oct 24 Yanping Chen Mathematical Sciences, UTD The Boltzmann collision operator for a
cylindrically symmetric velocity distribution function in a plasma

We develop a model for collision processes in industrially relevant plasmas. To reduce the computational cost of solving the Maxwell-Boltzmann equations, we assume that the velocity distribution function is cylindrically symmetric in velocity space and only axially dependent in physical space. Then the Boltzmann collision operator is also cylindrically symmetric. If the external force is only axially dependent, the Maxwell-Boltzmann system reduces to a system
of equations in two velocity and one spatial dimensions.
Oct 17 Brian Brennan Mathematics, Baylor U. Numerical Analysis of a Multi-Physics Model for Trace Gas Sensors Abstract
Oct 10 Dan Reynolds Mathematics, SMU Scalable Algorithms for Multi-physics Simulations: Modeling Cosmological Reionization Since the scientific revolution, the desire to create realistic models for physical systems has provided a continual stream for mathematical research. With the advent of computing these models have only increased in both accuracy and complexity, enabling detailed study of intricate physical processes. At the same time, however, as these models grow more detailed they increasingly exhaust the limits of classical numerical methods, posing new challenges in computational mathematics research.

In this talk, I will discuss recent research into numerical methods for large scale multi-physics simulations. Within this broad field, our work focuses on scalable algorithms for implicit-time simulations of coupled systems of partial differential equations. We primarily consider systems of nonlinear PDEs, that model physical processes that evolve on significantly different time scales. In this talk we will examine these issues in the context of astrophysical simulations for cosmological reionization within the early universe, although these mathematical issues and methods equally apply to a broad range of computational problems in science and engineering.
Oct 3 Panel Discussion Mathematical Sciences, UTD Interactive Engagement in Mathematics Courses Interactive Engagement is a pedagogical method that promotes conceptual understanding of students using classroom activities that yield immediate feedback through discussion with peers and
instructors. A recent NSF-sponsored study has demonstrated the effectiveness of interactive engagement in calculus courses. In this special seminar, a panel of graduate students and faculty will share their experiences employing interactive engagement and related active learning methods in Lectures and Recitation Sections here at UTD.

Graduate student Teaching Assistants and all those teaching freshman and sophomore Mathematics courses are strongly encouraged to attend and participate in the discussion.
Sep 26 Marcio Borges National Laboratory for Scientific Computing,
Markov chain Monte Carlo methods (MCMC) applied to porous media flows Abstract
Sep 12 Jordan Kaderli Mathematical Sciences, UTD Investigation of Microseismic Source Location via Full Waveform Inversion
Sep 5 Panel of Graduate Students Mathematical Sciences, UTD What I did this summer In the first Computational Science Seminar of the year, several mathematics and statistics graduate students will share their experiences doing an internship or attending a conference or workshop this summer. Students will talk about the application process, what they did, and what they learned from their experience. You are encouraged to ask questions and also to share your own experiences.