Wednesdays at 4:15 PM in 384H.
The Cucker-Smale systems is a popular model of collective behavior of interacting agents, used, in particular, to model bird flocking and fish swarming. The underlying premise is the local alignment of the bird (or fish, or any other agent) velocities. The Euler-Cucker-Smale system is an effective macroscopic PDE limit of such particle systems. It has the form of the pressureless Euler equations with a non-linear density-dependent alignment term. The alignment term is a non-linear version of the fractional Laplacian to a power $\alpha\in(0,1)$. It is known that the corresponding Burgers' equation with a linear dissipation of this type develops shocks in a finite time. We show that nonlinearity enhances the dissipation, and the solutions stay globally regular for all $\alpha\in(0,1)$: the dynamics is regularized due to the nonlinear nature of the alignment.
We first review G-equation (a level set Hamilton-Jacobi equation modeling interface motion) and front speeds in two dimensional cellular flow, then discuss front speeds in three-dimensional Arnold-Beltrami-Childress (ABC) flow. The ABC flow can be viewed as a perturbation of cellular flow, or a nearly integrable system when one of the ABC parameters is small. We show the existence of ballistic orbits where at least one component grows linearly in time, based on the Hamiltonian formulation and time reversal symmetries of the flows. The symmetry based construction works in the non-perturbative case when all three ABC parameters are equal to one. The linear growth law of front speeds of G-equation follows from the ballistic orbits. We end with open problems of front speeds in ABC flows for dissipative G-equations and reaction-diffusion-advection equations.
Given the solution u of an elliptic PDE, there is a naturally associated parabolic PDE for which u is a fixed point in time. We propose the application of parabolic techniques (Brownian motion via the stochastic interpretation) to study the elliptic equation; most of the talk will be spent on a variety of applications using this underlying philosophy and include a geometric inequality (improving on a classical result of Elliot Lieb), capacity interpretations of dimensionality reduction techniques with applications in big data, localization phenomena of Schrodinger operators and probabilistic techniques for bounding eigenvalues.
Surface motility such as swarming is thought to precede biofilm formation during infection. Population of bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. This collective mechanism of cell-cell coordination was shown to moderate swarming direction of individual bacteria to avoid antibiotics. In the second half of the talk, a novel three-dimensional multi-scale model will be described for simulating receptor-mediated adhesion of deformable platelets at the site of vascular injury under different shear rates of blood flow. The modeling approach couples submodels at three biological scales crucial for the early clot formation: novel hybrid cell membrane submodel to represent physiological elastic properties of a platelet, stochastic receptor–ligand binding submodel to describe cell adhesion kinetics and lattice Boltzmann submodel for simulating blood flow. Predictive model simulations revealed that platelet deformation, interactions between platelets in the vicinity of the vessel wall as well as the number of functional GPIbα platelet receptors played significant roles in platelet adhesion to the injury site. Lastly, macro- scale model of blood clot deformation and deformation of blood vessel will be described. The model represents clot as a ternary complex fluid made of heterogeneously distributed platelets, fibrin network and plasma phases and takes into account mechanical properties of different phases.
The sparsity structure of a system of polynomial equations or an optimization problem can be naturally described by a graph summarizing the interactions among the decision variables. It is natural to wonder whether the structure of this graph might help in computational algebraic geometry tasks (e.g., in solving the system). In this talk we will provide an introduction to this area, focused on the key notions of chordality and treewidth, which are of great importance in related areas such as numerical linear algebra, database theory, constraint satisfaction, and graphical models. In particular, we will discuss “chordal networks”, a novel representation of structured polynomial systems that provides a computationally convenient decomposition of a polynomial ideal into simpler (triangular) polynomial sets, while maintaining its underlying graphical structure. As we will illustrate through examples from different application domains, algorithms based on chordal networks can significantly outperform existing techniques. Based on joint work with Diego Cifuentes (MIT).
We describe a new class of computational optical sensors and imagers that do not rely on traditional refractive or reflective focusing but instead on special diffractive optical elements integrated with CMOS photodiode arrays. The diffractive elements have provably optimal optical properties essential for imaging, and act as a visual chirp and preserve full Fourier image information on the photodiode arrays. Images are not captured, as in traditional imaging systems, but rather computed from raw photodiode signals. Because such imagers forgo the use of lenses, they can be made unprecedentedly small—as small as the cross-section of a human hair. Such imagers have extended depth of field, from roughly 1mm to infinity, and should find use in numerous applications, from endoscopy to infra-red and surveillance imaging and more. Furthermore, the gratings and signal processing can be tailored to specific applications from visual motion estimation to barcode reading and others.
The Green's function offers a useful description to the electronic structure models, alternative to using eigenfunctions of the Hamiltonian operator. In this talk, we will demonstrate the usefulness of the Green's function perspective by two recent results: A mathematical analysis of the divide-and-conquer method and a new Green's function embedding approach PEXSI-\Sigma.
Consider a compressible fluid that is subject to a standing acoustic wave. Particles within the fluid are subject to an acoustic radiation force and tend to move to minima of the associated potential. In many cases, these minima coincide with the nodal sets of the standing wave, which are solutions to the Helmholtz equation. We present two methods for solving the control problem of finding the settings for transducers lining a reservoir (i.e. boundary conditions), necessary to best approximate a desired particle pattern within the reservoir. In the first method, a discretized version of the control problem is reduced to finding the smallest eigenvalue of a matrix. In the second method, we use an approximation result of functions by entire solutions to the Helmholtz equation to give an efficient and explicit solution to the control problem. An application of this principle is to fabricate selectively reinforced composite materials, where the matrix is a photo-cured polymer and the inclusions are carbon nanotubes.
Branching Brownian motion (BBM) is a classical process in probability, describing a population of particles performing independent Brownian motion and branching according to a Galton Watson process. Arguin et al. and Aidekon et al. proved the convergence of the extremal process. In the talk we discuss how one can obtain finer results on the extremal level sets by using a random walk-like representation of the extremal particles. We establish among others the asymptotic density of extremal particles at a given distance from the maximum and the upper tail probabilities for the distance between the maximum and the second maximum (joint work with Aser Cortines and Oren Louidor).
The Fock operator, which appears in the widely used Hartree-Fock theory and Kohn-Sham density functional theory with hybrid exchange-correlation functionals, plays a central role modern quantum chemistry and materials science. The computational cost associated with the Fock exchange operator is however very high. In a simplified setting, the Hartree-Fock equation requires the computation of low-lying eigenpairs of a large matrix in the form A+B. Here applying A to a vector is easy but A has a large spectral radius, while applying B (the Fock operator) is costly but B has a small spectral radius. It turns out that most eigensolvers are not well equipped to solve such problems efficiently. We develop an adaptive compressed method to efficiently treat such eigenvalue problems. We prove that the method converges locally, and surprisingly, converges globally from almost everywhere. The adaptive compression method has been adopted in community electronic structure software packages such as Quantum ESPRESSO, and offers an order of magnitude speedup compared to existing methods. We also demonstrate that the adaptive compression method can enable hybrid functional calculations in planewave basis sets with more than 4000 atoms, and discuss the possible applications of the adaptive compression method beyond the Hartree-Fock-like equations.
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