CSCAMM Archives for Academic Year 2014


CSCAMM Seminar: Mixing and enhanced dissipation in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow

When: Wed, September 3, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Jacob Bedrossian, CSCAMM and the Department of Mathematics, UMD
Abstract: In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times t > Re^(1/3), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by a mixing-enhanced dissipation effect. Afterward, the remaining shear flow decays on very long time scales t >> Re back to the Couette flow. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re^(-1)) L2 function.

Joint with Nader Masmoudi and Vlad Vicol.

CSCAMM Seminar: Asymmetric Auctions

When: Wed, September 10, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Gadi Fibich, Tel Aviv University
Abstract: Most of auction theory concerns the case where all bidders are symmetric (identical). This is not because bidders are believed to be symmetric, but rather because the analysis of asymmetric auctions is considerably harder. For example, in the case of the common first-price auction, the symmetric case is governed by a single ODE which is easy to solve explicitly. In contrast, the model for asymmetric first-price auction consists of n first-order nonlinearly coupled ODES with 2n boundary condition and an unknown location of the right boundary, where n is the number of bidders. This nonstandard boundary value problem is challenging to analyze, or even to solve numerically. Therefore, very little is known about its solutions.

In this talk I will review various approaches to this problem (perturbation analysis, dynamical systems, numerical methods). I will mainly focus on some recent results for the case when the number of bidders is large (n>>1). In this case the solution develops a nonstandard boundary layer structure, and one can obtain a surprising revenue equivalence result between asymmetric first-price and second-price auctions.

This is a joint work with Nir Gavish and Arieh Gavious

CSCAMM Seminar: The Huygens' principle as a computational tool

When: Wed, September 17, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Semyon V. Tsynkov, North Carolina State University
Abstract: We will discuss the (strong) Huygens' principle as it applies to Maxwellian electrodynamics, and show how it can be exploited for the design of numerical methods with specific advantageous properties. In particular, we will demonstrate that it can help one obtain temporally uniform error bounds over arbitrarily long time intervals. Theoretical developments will be corroborated by numerical simulations, including those performed using third party production CEM and/or plasma codes.

Collaborators: V. Ryaben'kii, V. Turchaninov, S. Petropavlovsky, and Computational Sciences, LLC. Funding: NSF, AFOSR, and ARO (STTR Phase I and II).

CSCAMM Seminar: An extended shallow water model with viscous layer

When: Wed, September 24, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Francois James, University of Orleans
Abstract: A classical way to obtain the shallow water model is by integrating the Navier-Stokes equations along the vertical. The closure of the system is ensured by assumptions on the vertical velocity profile. For instance a constant velocity profile corresponds to a perfect fluid. This leads to a model which does not take into account the friction of the fluid on the bottom. Empirical friction laws can be added afterwards. We propose here an approximation which takes into account in a more detailed way the viscous layer above the bed. This leads to an extended shallow water model with a friction term and some correction in the pressure terms as well. We shall derive in some details the set of equations and give some numerical illustrations.

Joint CSCAMM/KI-Net Seminar: UQ Algorithms for Extreme-scale Systems

When: Wed, October 1, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Dongbin Xiu, Department of Mathematics, and Scientific Computing and Imaging Institute, University of Utah

Abstract: The field of uncertainty quantification (UQ) has received an increasing amount of attention recently. Extensive research efforts have been devoted to it and many novel numerical techniques have been developed. These techniques aim to conduct stochastic simulations for very large-scale complex systems. Although remarkable progresses have been made, UQ simulations remains challenging due to their exceedingly high simulation cost for problems at extreme scales. In this talk I will discuss some of the recent developed UQ algorithms that are particularly suitable for extreme-scale simulations. These methods are (1) collocation-based, such that they can be directly applied to systems with legacy simulation codes; and (2) capacity-based, such that they deliver the (near) optimal simulation accuracy based on the available simulation capacity. In another word, these methods deliver the best UQ simulation results based on any given computational resource one can afford, which is often very limited at the extreme scales.

Joint CSCAMM/KI-Net Seminar: Interfases and flux-saturated mechanisms

When: Wed, October 15, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Juan Soler, Department of Applied Mathematics, University of Granada

Abstract: The aim of this talk is to provide a qualitative analysis of flux saturated operators in one dimension: regularity and regularizing effects, dynamics of discontinuous interfaces, existence of traveling wave profiles, speed and waiting time for the growth of the support, deduction from first principles (optimal mass transport, hydrodynamic limit of kinetic equations or nonlinear Hilbert expansion)... The goal is to better understand and characterize these phenomenona by focusing on two prototypical operators: the relativistic heat equation and the flux-limited porous media equation. In the treatment of interfaces we show that flux saturated models behave more closely to conservation laws than to diffusive models. Finally, some applications are shown in developmental biology.

CSCAMM Seminar: Laplacian multi-wavelets bases for high-dimensional data and their applications

When: Mon, October 20, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Nir Sharon, Department of Applied Mathematics, Tel-Aviv University
Abstract: We introduce a framework for representing functions dened on high-dimensional data. In this framework, we propose to use the eigenvectors of the graph Laplacian to construct a multiresolution analysis on the data, results in a one parameter family of orthogonal bases. We describe the construction of such basis, its properties and derive a bound on the decay rates of the expansion coecients. In addition, the question of measuring the smoothness of discrete functions is addressed based on a discrete analogue of Besov spaces. We also present a few applications for this family of bases and report an ongoing research related to future applications.

This is a joint work with Yoel Shkolnisky.

CSCAMM Seminar: TBA

When: Wed, October 29, 2014 - 2:00pm
Where: CSIC #4122
Speaker: TBA
Abstract: TBA

CSCAMM Seminar: A constraint-preserving numerical method for approximating wave maps into the sphere

When: Wed, November 5, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Franziska Weber, University of Oslo, Department of Mathematics
Abstract: We present a convergent finite difference method for approximating wave maps into the sphere. The method is based on a reformulation of the second order equation as a first order system by using the angular momentum as an auxilary variable. This enables us to preserve the length constraint as well as the energy at the discrete level. The method is shown to converge to a weak solution of the wave map equation as the discretization parameters go to zero. It is fast in the sense that O(N log N) operations are required in each timestep (where N is the number of grid cells) and a linear CFL-condition is suffcient for stability and convergence. The performance of the method is illustrated by numerical experiments.

CSCAMM Seminar: Control strategies for interacting particle systems

When: Wed, November 19, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Michael Herty, Department of Mathematics, RWTH Aachen University

Abstract: We present a control approach for large systems of interacting multi--agents based on the Riccati equation. If the agent dynamics enjoys a strong symmetry the arising high dimensional Riccati equation is simplified and the resulting coupled system allows for a formal mean--field limit. In case of linear dynamics and quadratic objective function the presented approach is optimal and is compared to the (suboptimal) model predictive control strategies. The relation to meanfield control and the Hamilton-Jacobi Bellmann equation is also discussed as well as the relation to recently introduced best-reply strategies in pedestrian and wealth dynamics.

CSCAMM Seminar: Phase Retrieval for single-shot lensless FEL experiment

When: Wed, December 3, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Dr. Oren Raz, Department of Chemistry & Biochemistry, University of Maryland

Abstract: Phase retrieval is a class of problems in which a signal is reconstructed from the measurements of the absolute values of its projections. These problems are common in many experimental setups, ranging from astronomy through pulse characterization to X-ray imaging. In my talk I will focus on a specific type of phase retrieval problems, motivated by recent advances in X-ray sources: reconstructing a single particle from its diffraction pattern amplitudes. I will show that although this problem is, in general, very difficult, it can be reduced to a surprisingly simple problem if there are two separated particles rather then a single one. I will demonstrate this on recently taken X-ray data sets, in which, by mistake, there were two rather then one particle.

Joint KI-Net/CSCAMM Seminar: The Nonlinear Schrodinger Equation and Weak Turbulence

When: Wed, December 10, 2014 - 2:00pm
Where: CSIC #4122
Speaker: Prof. Pierre Germain, Courant Institute of Mathematical Sciences, NYU

Abstract: The theory of weak turbulence has been put forward by applied mathematicians to describe the asymptotic behavior of NLS set on a compact domain - and of many other infinite dimensional systems. It is believed to be valid in a statistical sense, in the weakly nonlinear, infinite volume limit. I will present how these limits can be taken rigorously, and give rise to new equations.

Intrinsic complexity of differential operators

When: Wed, January 28, 2015 - 2:00pm
Where: CSCAMM 4122
Speaker: Hongkai Zhao (University of California - Irvine) - http://www.math.uci.edu/~zhao/homepage/home/home.html
Abstract: Approximate separable representation of the Green’s functions for differential operators (with appropriate boundary conditions) is a basic and important question in analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity or degrees of freedom of the corresponding differential equation. Computationally, being able to approximate a Green’s function as a sum with few separable terms is equivalent to the existence of low rank approximation of the corresponding discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, two types of differential operators will be discussed. One is coercive elliptic equation in divergence form, which is highly separable. The other one is Helmholtz equation in high frequency limit for which we developed a new approach to study the approximate separability of Green’s function based on an geometric characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness and implications for computation setups that are commonly used in practice. This is a joint work with Bjorn Engquist.

Phantom jams and jamitons in macroscopic traffic models

When: Wed, February 4, 2015 - 2:00pm
Where: CSIC Bldg 4122
Speaker: Benjamin Seibold (Department of Mathematics, Temple University) - http://math.temple.edu/~seibold/
Abstract: Initially homogeneous vehicular traffic flow can become inhomogeneous even in the absence of obstacles. Such ``phantom traffic jams'' can be explained as instabilities of a wide class of ``second-order'' macroscopic traffic models. In this unstable regime, small perturbations amplify and grow into nonlinear traveling waves. These traffic waves, called ``jamitons'', are observed in reality and have been reproduced experimentally. We show that jamitons are analogs of detonation waves in reacting gas dynamics, thus creating an interesting link between traffic flow, combustion, water roll waves, and black holes. This analogy enables us to employ the Zel'dovich-von Neumann-Doering theory to predict the shape and travel velocity of the jamitons. We furthermore demonstrate that the existence of jamiton solutions can serve as an explanation for multi-valued parts that fundamental diagrams of traffic flow are observed to exhibit.

Global well-posedness and blow up for active scalar equations

When: Wed, February 11, 2015 - 2:00pm
Where: CSIC Bldg 4122
Speaker: Dr. Michele Coti Zelati (Department of Mathematics, University of Maryland) - http://www.math.umd.edu/~micotize/
Abstract: We aim to study the behavior of solutions to a class of active scalar equations, for which the two-dimensional surface quasi-geostrophic and the Burgers equations are the main instances. When (nonlocal) dissipation is present, different scenarios may occur depending on the dissipative operator: in the Burgers equation, the picture is fairly clear, while for the SQG equation the global existence of regular solutions in the supercritical (i.e. weakly dissipative) regime is an outstanding open problem. We prove that the SQG equation in the supercritical case continuously depends on the dissipative operator, achieving regularity for large initial data in critical spaces. Time permitting, we will discuss possible different (inviscid) regularizations of such equations that lead to global regularity or blow-up of solutions in a similar fashion.

Continuum limit of total variation on point clouds

When: Wed, February 25, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Nicolas Garcia Trillos (Department of Mathematical Sciences, Carnegie Mellon University) -
Abstract: We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We study when is the cut capacity, and more generally total variation, on these graphs a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. Applications to the study of consistency of cut based clustering procedures will be discussed.

The mathematics of three-dimensional structure determination of molecules by cryo-electron microscopy

When: Wed, March 11, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Amit Singer (Department of Mathematics and PACM, Princeton University) - https://web.math.princeton.edu/~amits/
Abstract: Cryo-electron microscopy (EM) is used to acquire noisy 2D projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal is to reconstruct the 3D structure of the macromolecule with sufficiently high resolution. We will discuss algorithms for solving the cryo-EM problem and their relation to other branches of mathematics such as tomography, random matrix theory, representation theory, convex optimization and semidefinite programming.

Stablized and Adaptive Time-Stepping Methods for Phase-Field Models

When: Wed, March 25, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Tao Tang (Department of Mathematics, Hong Kong Baptist University) - http://www.math.hkbu.edu.hk/~ttang/
Abstract: Recent results in the literature provide computational evidence that stabilized time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear diffusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the derivative of the nonlinear potential function satisfies a Lipschitz type condition. This talk will discuss how to remove the Lipschitz assumption on the nonlinearity. On the numerical algorithm side, we will demonstrate how to design high-order and adaptive methods for the phase-dield equations.

Spectral Difference Method for Computational Fluid Dynamics : From Flapping/Rotary Wing Aerodynamics to Thermal Convection of the Sun

When: Wed, April 8, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Chunlei Liang (George Washington University) - http://www.mae.seas.gwu.edu/chunlei-liang
Abstract: Since the beginning of the 21st century, high-order methods for Computational Fluid Dynamics (CFD) have been quickly adopted by the engineering community for various flow problems because they can produce accurate results on relatively coarse grids. The speaker will focus on the advancement of the high-order spectral difference method (SDM) for solving compressible Naiver-Stokes type equations on unstructured grids. The first part of this lecture is about studies of flapping wing aerodynamics which have demonstrated the suitability of SDM for predicting unsteady vortex dominated flow on moving and deforming domains. The second part of this lecture will discuss a simple, novel, and high-order curved sliding interface method to enable the SDM for simulating unsteady flow on coupled stationary and rotary domains for rotary wing aerodynamics. Finally, the SDM is further advanced for predicting stratified thermal convection in the sun. The speaker will report a successful effort for building a Compressible High-ORder Unstructured Spectral-difference (CHORUS) code for predicting the convection zone of the sun in collaboration with the National Center for Atmospheric Research (NCAR).

Necklace solitary waves on bounded domains

When: Wed, April 15, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Gadi Fibich (Department of Applied Mathematics, Tel Aviv University) - http://www.math.tau.ac.il/~fibich/
Abstract: In this talk I will present new solitary waves of the two-dimensional nonlinear Schrodinger equation on bounded domains, which have a "necklace" structure. I will consider their structure, stability, and how to compute them.

Computational Modeling of the Eukaryotic Cytoskeleton

When: Wed, April 29, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Garegin Papoian (Department of Chemistry and Biochemistry, University of Maryland) - http://papoian.chem.umd.edu
Abstract: Cells of higher organisms contain a dynamically remodeling filamentous network, called cytoskeleton, comprised of actin, myosin and many other molecules. The cytoskeletons endue cells with their instantaneous shapes, providing a machinery for cells to move around, generate forces and also integrate both chemical and mechanical signaling. In terms of the physico-chemical mechanisms, the underlying acto-myosin network growth and remodeling processes are based on a large number of chemical and mechanical interactions, which are mutually coupled, and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed a detailed physico-chemical, stochastic model of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work shedded light on the complex, non-linear feedbacks between the chemical and mechanical processes governing actomyosin network dynamics, highlighting, in particular, the importance of diffusional and active transport phenomena.

The atomistic origin of crystal growth: Study in 1+1 dimensions

When: Wed, May 6, 2015 - 2:00pm
Where: CSIC 4122
Speaker: Prof. Dionisios Margetis (Department of Mathematics, University of Maryland) - http://www.math.umd.edu/~diom/
Abstract: Crystals are ubiquitous in nature and in manmade devices. The growth of crystals at sufficiently low temperatures (e.g., room temperature) is characterized by the existence of certain defects. "Mesoscale" theories for the motion of such defects have been proposed since the 1950s but they are largely phenomenological. In this talk, I discuss the derivation of a mesoscale theory of crystals defects (steps) from a kinetic atomistic perspective.