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ACMS Abstracts: Spring 2014

Adrianna Gillman (Dartmouth)

Fast direct solvers for linear partial differential equations

The cost of solving a large linear system often determines what can and cannot be modeled computationally in many areas of science and engineering. Unlike Gaussian elimination which scales cubically with the respect to the number of unknowns, fast direct solvers construct an inverse of a linear in system with a cost that scales linearly or nearly linearly. The fast direct solvers presented in this talk are designed for the linear systems arising from the discretization of linear partial differential equations. These methods are more robust, versatile and stable than iterative schemes. Since an inverse is computed, additional right-hand sides can be processed rapidly. The talk will give the audience a brief introduction to the core ideas, an overview of recent advancements, and it will conclude with a sampling of challenging application examples including the scattering of waves.

Yaniv Plan (Michigan)

Low-dimensionality in mathematical signal processing

Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.

Lorenzo Pareschi (University of Ferrara)

Kinetic description and simulation of optimal control problems in self-organized systems

Emerging phenomena driven by interactions of a large number of self-organized agents are present in various real life applications. Different to the classical approach where individuals are assumed to freely interact with each other, here we are particularly interested in such problems in a constrained setting. This can be used to understand the influence of external factors to the system dynamics, for example to enforce emergence of non spontaneous desired asymptotic states. Classical examples are given by persuading voters to vote for a specific candidate, by influencing buyers towards a given good or asset or by confining/driving a group of animals in a specific area. In this talk we review different kind of controls for the resulting process and present several kinetic models and stochastic simulation methods including those controls.

Mohammed Lemou (Institut de Recherche Mathématique de Rennes)

Quantitative stability inequalities for Vlasov-Poisson and 2d Euler systems

In this work, we prove a new functional inequality of Hardy-Littlewood type for generalized rearrangements of functions. We then show how this inequality yields a quantitative stability result for dynamical systems that essentially have the important property to preserve the rearrangement and the Hamiltonian. In particular we derive a quantitative stability result for a large class of steady state solutions to Vlasov-Poisson systems by providing a quantitative control of the L1 norm of the perturbation (uniformly in time) by the perturbed Hamiltonian and the L1 norm of the perturbation at initial time. In fact, such nonlinear stability has already been recently obtained, but our proof was based in a crucial way on compactness arguments which by construction provide no quantitative control of the perturbation. We finally investigate the application of our Hardy-Littlewood type inequality to other contexts such as the relativistic Vlasov-Poisson and 2D-Euler systems.

Harvey Segur (Colorado)

The nonlinear Schrödinger equation, dissipation and ocean swell

The focus of this talk is less about how to solve a particular mathematical model, and more about how to find the right model of a physical problem.

The nonlinear Schrödinger (NLS) equation was discovered as an approximate model of wave propagation in several branches of physics in the 1960s. It has become one of the most studied models in mathematical physics, because of its interesting mathematical structure and because of its wide applicability – it arises naturally as an approximate model of surface water waves, nonlinear optics, Bose-Einstein condensates and plasma physics.

In every physical application, the derivation of NLS requires that one neglect the (small) dissipation that exists in the physical problem. But our studies of water waves (including freely propagating ocean waves, called “ocean swell”) have shown that even though dissipation is small, neglecting it can give qualitatively incorrect results. This talk describes an ongoing quest to find an appropriate generalization of NLS that correctly predicts experimental data for ocean swell. As will be shown, adding a dissipative term to the usual NLS model gives correct predictions in some situations. In other situations, both NLS and dissipative NLS give incorrect predictions, and the “right model” is still to be found.

This is joint work with Diane Henderson, at Penn State.

Elena D'Onghia (UW)

The origin of spiral arms in galactic disks

The precise nature of spiral structure in galaxies remains uncertain. Using high-resolution N-body simulations, I follow the motions of stars under the influence of gravity, and show that mass concentrations with properties similar to those of giant molecular clouds or clumps of gas in the galactic disk can induce the development of spiral arms through a process termed "swing amplification". However, unlike in earlier work, I will demonstrate that the eventual response of the disk is highly non-linear, significantly modifying the formation and longevity of the resulting patterns. I will discuss how these findings affect phenomena occurring in the stellar disk, like the migration of the Sun from its birth place.