Prof. Jacob Bedrossian, CSCAMM and the Department of Mathematics, University of Maryland
Mixing and enhanced dissipation in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
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), and in particular exhibits "inviscid damping" (e.g. the vorticity mixes and weakly approaches a shear flow). 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.
Prof. Gadi Fibich, CSCAMM, University of Maryland and Department of Applied Mathematics, Tel-Aviv University
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
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).
An extended shallow water model with viscous layer
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.
Prof. Dongbin Xiu, Department of Mathematics, and
Scientific Computing and Imaging Institute, University of Utah
UQ Algorithms for Extreme-scale Systems
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.
Parallelizable Block Iterative Methods for Stochastic Processes
In many applications involving large systems of stochastic differential equations, the states space can be partitioned into groups which are only weakly interacting. For example, molecular dynamics simulations of large molecules undergoing Langevin dynamics may be divided into smaller components, each at equilibrium. If the components are decoupled, then the equilibrium distribution of the entire system is a product of the marginals and can be computed in parallel. However, taking interactions into account, the entire state of the system must be considered as a whole and na´ve parallelization is not possible. We propose an iterative method along the lines of the wave-form relaxation approach for calculating all component marginals. The method allows some parallelization between conditionally independent components, depending on the minimal coloring of the graph describing their mutual interactions. Joint work with Ben Leimkuhler and Matthias Sachs (University of Edinburgh).
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.
Nir Sharon, Department of Applied Mathematics, Tel-Aviv University
Laplacian multi-wavelets bases for high-dimensional data and their applications
We introduce a framework for representing functions defined 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 coefficients. 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.
A constraint-preserving numerical method for approximating wave maps into the sphere
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.
Control strategies for interacting particle systems
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.
Dr. Oren Raz, Department of Chemistry & Biochemistry, University of Maryland
Phase Retrieval for single-shot lensless FEL experiment
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.
The Nonlinear Schrodinger Equation and Weak Turbulence
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.