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.
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.
Sangtae Kim (Purdue)
The Faxén Laws of Stokes flow and their connection to singularity solutions
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.
Michael Shelley (Applied Math Lab, Courant Institute, NYU)
Mathematical models of soft active materials
Soft materials that have an "active" microstructure are important examples of so-called active matter. Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation.
Paul Hand (MIT)
Evaluating signal recovery algorithms with semirandom models
The planted clique and sparse principal component analysis problems involve identifying a specific ordered structure within a noisy environment. There are many algorithms for such tasks, and it is important to have a theoretical understanding of which algorithms are better for "typical" problem instances. The simplest environment for evaluating these algorithms is a planted-random model, where the signal is buried in some uniform noise. This type of model is often not typical enough, as it affords comparable performance for some robust and non-robust algorithms. Thus, there is a need for more sophisticated models. Semirandom models are like planted-random models, but they allow adversarial alteration. The semirandom model prevents methods from exploiting distribution-specific properties that are likely not true in typical problem instances. Thus, they provide a reasonable context in which to evaluate algorithm performance under a robustness requirement. In this talk, we will present existing results for planted clique with planted random and semirandom models. We will also discuss the potential for semirandom models in two other problems: sparse principal component analysis and the problem of finding the sparsest element in a subspace.
Scott Hottovy (UW)
Modeling with stochastic differential equations: A noisy circuit and thresholds for rainfall
In this talk I will discuss modeling physical systems using stochastic differential equations (SDE) with two specific examples: a noisy electrical circuit and the onset of convection in clouds. SDE are used to model many systems governed by deterministic dynamics with intrinsic noise. For some models, when the noise depends on the state of the system, the SDE depends on the construction of the stochastic integral (Itô, Stratonovich). For example, in a population model, the choice of stochastic integral will lead to growth in the Stratonovich construction but extinction in the Itô choice. I will discuss an interesting experiment of an electrical circuit that shifts the SDE model from obeying Stratonovich calculus to obeying Itô calculus.
In the second part of this talk, I will describe ongoing work with Sam Stechmann on modeling moisture in a one column cloud. Interesting statistics can be solved for exactly by using a simple SDE model, Brownian motion with drift, where the dynamics change when a threshold is reached. I will compare and contrast four different models of the transition to rainfall using random and deterministic thresholds and show that a simplified model can be used as an approximation in some regimes.