Seminars and Colloquia by Series

Matrix Perturbation and Manifold-based Dimension Reduction.

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Xiaoming Huo Georgia Tech (School of ISyE)
Many algorithms were proposed in the past ten years on utilizing manifold structure for dimension reduction. Interestingly, many algorithms ended up with computing for eigen-subspaces. Applying theorems from matrix perturbation, we study the consistency and rate of convergence of some manifold-based learning algorithm. In particular, we studied local tangent space alignment (Zhang & Zha 2004) and give a worst-case upper bound on its performance. Some conjectures on the rate of convergence are made. It's a joint work with a former student, Andrew Smith.

Multiscale modeling of granular flow

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Chris RycroftUC-Berkeley
Due to an incomplete picture of the underlying physics, the simulation of dense granular flow remains a difficult computational challenge. Currently, modeling in practical and industrial situations would typically be carried out by using the Discrete-Element Method (DEM), individually simulating particles according to Newton's Laws. The contact models in these simulations are stiff and require very small timesteps to integrate accurately, meaning that even relatively small problems require days or weeks to run on a parallel computer. These brute-force approaches often provide little insight into the relevant collective physics, and they are infeasible for applications in real-time process control, or in optimization, where there is a need to run many different configurations much more rapidly. Based upon a number of recent theoretical advances, a general multiscale simulation technique for dense granular flow will be presented, that couples a macroscopic continuum theory to a discrete microscopic mechanism for particle motion. The technique can be applied to arbitrary slow, dense granular flows, and can reproduce similar flow fields and microscopic packing structure estimates as in DEM. Since forces and stress are coarse-grained, the simulation technique runs two to three orders of magnitude faster than conventional DEM. A particular strength is the ability to capture particle diffusion, allowing for the optimization of granular mixing, by running an ensemble of different possible configurations.

Fast algorithms for the computation of the pseudospectral abscissa and pseudospectral radius.

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Nicola GuglielmiUniversità di L'Aquila
This is a joint work with Michael Overton (Courant Institute, NYU). The epsilon-pseudospectral abscissa and radius of an n x n matrix are respectively the maximum real part and the maximal modulus of points in its epsilon-pseudospectrum. Existing techniques compute these quantities accurately but the cost is multiple SVDs of order n, which makesthe method suitable to middle size problems. We present a novel approach based on computing only the spectral abscissa or radius or a sequence of matrices, generating a monotonic sequence of lower bounds which, in many but not all cases, converges to the pseudospectral abscissa or radius.

Mathematical Paradigms for Periodic Phase Separation and Self-Assembly of Diblock Copolymers

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2009 - 13:00 for 30 minutes
Location
Skiles 255
Speaker
Rustum ChoksiSimon Fraser University

Please Note: A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations, I will attempt to characterize minimizers without any preassigned bias for their geometry.

Energy-driven pattern formation induced by competing short and long-range interactions is ubiquitous in science, and provides a source of many challenging problems in nonlinear analysis. One example is self-assembly of diblock copolymers. Phase separation of the distinct but bonded chains in dibock copolymers gives rise to an amazingly rich class of nanostructures which allow for the synthesis of materials with tailor made mechanical, chemical and electrical properties. Thus one of the main challenges is to describe and predict the self-assembled nanostructure given a set of material parameters.

A spectral method with window technique for the initial value problems of the Kadomtsev-Petviashvili equation

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 26, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Chiu-Yen Kao Ohio State University (Department of Mathematics)
The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersivewave equation which was proposed to study the stability of one solitonsolution of the KdV equation under the influence of weak transversalperturbations. It is well know that some closed-form solutions can beobtained by function which have a Wronskian determinant form. It is ofinterest to study KP with an arbitrary initial condition and see whetherthe solution converges to any closed-form solution asymptotically. Toreveal the answer to this question both numerically and theoretically, weconsider different types of initial conditions, including one-linesoliton, V-shape wave and cross-shape wave, and investigate the behaviorof solutions asymptotically. We provides a detail description ofclassification on the results. The challenge of numerical approach comes from the unbounded domain andunvanished solutions in the infinity. In order to do numerical computationon the finite domain, boundary conditions need to be imposed carefully.Due to the non-periodic boundary conditions, the standard spectral methodwith Fourier methods involving trigonometric polynomials cannot be used.We proposed a new spectral method with a window technique which will makethe boundary condition periodic and allow the usage of the classicalapproach. We demonstrate the robustness and efficiency of our methodsthrough numerous simulations.

Normal Mode Analysis for Drifter Data Assimilation to Improve Trajectory Predictions from Ocean Models

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 19, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Helga S. HuntleyUniversity of Delaware
Biologists tracking crab larvae, engineers designing pollution mitigation strategies, and climate scientists studying tracer transport in the oceans are among many who rely on trajectory predictions from ocean models. State-of-the-art models have been shown to produce reliable velocity forecasts for 48-72 hours, yet the predictability horizon for trajectories and related Lagrangian quantities remains significantly shorter. We investigate the potential for decreasing Lagrangian prediction errors by applying a constrained normal mode analysis (NMA) to blend drifter observations with model velocity fields. The properties of an unconstrained NMA and the effects of parameter choices are discussed. The constrained NMA technique is initially presented in a perfect model simulation, where the “true” velocity field is known and the resulting error can be directly assessed. Finally, we will show results from a recent experiment in the East Asia Sea, where real observations were assimilated into operational ocean model hindcasts.

[Special day and location] Scaling properties and suppression of Fermi acceleration in time dependent billiards

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, October 14, 2009 - 13:00 for 8 hours (full day)
Location
Skiles 269
Speaker
Edson Denis LeonelUniversidade Estadual Paulista, Rio Claro, Brazil
Fermi acceleration is a phenomenon where a classical particle canacquires unlimited energy upon collisions with a heavy moving wall. Inthis talk, I will make a short review for the one-dimensional Fermiaccelerator models and discuss some scaling properties for them. Inparticular, when inelastic collisions of the particle with the boundaryare taken into account, suppression of Fermi acceleration is observed.I will give an example of a two dimensional time-dependent billiardwhere such a suppression also happens.

A fast and exact algorithm of minimizing the Rudin-Osher-Fatemi functional in one dimension

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 12, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Wei ZhuUniversity of Alabama (Department of Mathematics)
The Rudin-Osher-Fatemi (ROF) model is one of the most powerful and popular models in image denoising. Despite its simple form, the ROF functional has proved to be nontrivial to minimize by conventional methods. The difficulty is mainly due to the nonlinearity and poor conditioning of the related problem. In this talk, I will focus on the minimization of the ROF functional in the one-dimensional case. I will present a new algorithm that arrives at the minimizer of the ROF functional fast and exactly. The proposed algorithm will be compared with the standard and popular gradient projection method in accuracy, efficiency and other aspects.

Biological aggregation patterns and the role of social interactions

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 28, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Chad TopazMacalester College
Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.

Adaptive spline interpolation: asymptotics of the error and construction of the partitions

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 21, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yuliya BabenkoDepartment of Mathematics and Statistics, Sam Houston State University
In this talk we first present the exact asymptotics of the optimal error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We further discuss the applications to numerical integration and adaptive mesh generation for finite element methods, and explore connections with the problem of approximating the convex bodies by polytopes. In addition, we provide the generalization to asymmetric norms. We give a brief review of known results and introduce a series of new ones. The proofs of these results lead to algorithms for the construction of asymptotically optimal sequences of triangulations for linear interpolation. Moreover, we derive similar results for other classes of splines and interpolation schemes, in particular for splines over rectangular partitions. Last but not least, we also discuss several multivariate generalizations.

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