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.
Seminars and Colloquia by Series
Monday, November 16, 2009 - 13:00 , Location: Skiles 255 , Chris Rycroft , UC-Berkeley , Organizer:
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.
Monday, November 9, 2009 - 13:00 , Location: Skiles 255 , Nicola Guglielmi , Università di L'Aquila , email@example.com , Organizer: Sung Ha Kang
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.
Monday, November 2, 2009 - 13:00 , Location: Skiles 255 , Rustum Choksi , Simon Fraser University , Organizer:
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 equationMonday, October 26, 2009 - 13:00 , Location: Skiles 255 , Chiu-Yen Kao , Ohio State University (Department of Mathematics) , firstname.lastname@example.org , Organizer: Sung Ha Kang
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 ModelsMonday, October 19, 2009 - 13:00 , Location: Skiles 255 , Helga S. Huntley , University of Delaware , Organizer:
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 billiardsWednesday, October 14, 2009 - 13:00 , Location: Skiles 269 , Edson Denis Leonel , Universidade Estadual Paulista, Rio Claro, Brazil , Organizer:
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.
Monday, October 12, 2009 - 13:00 , Location: Skiles 255 , Wei Zhu , University of Alabama (Department of Mathematics) , email@example.com , Organizer: Sung Ha Kang
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.
Monday, September 28, 2009 - 13:00 , Location: Skiles 255 , Chad Topaz , Macalester College , Organizer:
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.
Monday, September 21, 2009 - 13:00 , Location: Skiles 255 , Yuliya Babenko , Department of Mathematics and Statistics, Sam Houston State University , Organizer: Doron Lubinsky
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.
Wednesday, September 9, 2009 - 13:00 , Location: Skiles 114 , Amy Novick-Cohen , Technion , Organizer: John McCuan
Grain boundaries within polycrystalline materials are known to be governed by motion by mean curvature. However, when the polycrystalline specimen is thin, such as in thin films, then the effects of the exterior surfaces start to play an important role. We consider two particularly simple geometries, an axi-symmetric geometry, and a half loop geometry which is often employed in making measurements of the kinetic coefficient in the motion by mean curvature equation, obtaining corrective terms which arise due to the coupling of grain boundaries to the exterior surface. Joint work with Anna Rotman, Arkady Vilenkin & Olga Zelekman-Smirin