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ACMS Abstracts: Fall 2017

Jinzi Mac Huang (Courant)

Sculpting of a dissolving body

In geology, dissolution in fluids leads to natural pattern formations. For example the Karst topography occurs when water dissolves limestone, and travertine terraces form as a balance of dissolution and precipitation. In this talk, we consider the shape dynamics of a soluble object immersed in water, with either external flow imposed or convective flow under gravity. We find that different flow configurations lead to different shape dynamics, for example a terminal self-similar shape emerges from dissolving in external flow, while fine scale patterns form when no external flow is imposed. We also find that under gravity, a dissolving body with initially smooth surface evolves into an increasingly sharp needle shape. A mathematical model predicts that a geometric shock forms at the tip of dissolved body, with the tip curvature becoming infinite in finite time.

Dongnam Ko (Seoul National Univ.)

On the emergence of local flocking phenomena in Cucker-Smale ensembles

Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.

Yingwei Wang (UW-Madison)

Introduction to Muntz Polynomial Approximation

In general, solutions to the Laplacian equation enjoy relatively high smoothness. However, they can exhibit singular behaviors at domain corners or points where boundary conditions change type. In this talk, I will focus on the mixed Dirichlet-Neumann boundary conditions for Laplacian equation, and discuss how singularities in this case adversely affect the accuracy and convergence rates of standard numerical methods. Then, starting from the celebrated Weierstrass theorem about polynomial approximation, I will describe the approximation theory related to the so-called Muntz polynomials, which can be viewed as a generalization of usual polynomials. Additionally, I will illustrate the idea of Muntz-Galerkin methods, and show that how they can overcome the difficulties to achieving high order accuracy for the problems with singularities.

Jianlin Xia (Purdue Univ.)

Fast Randomized Direct Solvers for Large Linear Systems

In this talk, we discuss how randomized techniques can be used in structured matrix compression, and in turn in solving large dense and sparse linear systems. It is known that randomized sampling can help compute approximate SVDs via matrix-vector products. Such randomized ideas have been applied to some structured matrices for the fast compression of off-diagonal blocks. This leads to randomized and even matrix-free direct solvers for large dense linear systems.

Furthermore, the techniques can be extended to sparse direct solvers, where randomization helps compress dense fill-in in the factorization into skinny matrix-vector products. This has a significant advantage over dense or structured fill-in used before, since the processing and propagation of the skinny products are much simpler. For some sparse discretized problems (often elliptic), the randomized sparse direct solvers can reach nearly O(n) complexity.

We also show how to control the approximation accuracy in randomized structured solution, and further prove the superior backward stability of these randomized methods. Part of the work is joint with Yuanzhe Xi.

Yuri Lvov (Rensselaer Polytechnic Institute)

Fermi Pasta Ulam Tsingou (FPUT) chain - new ideas about old problem

Fermi-Pasta-Ulam-Tsingou chain is a theoretical model of a one dimensional crystal. It consists of point masses connected by nonlinear strings. Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted numerical experiments on this model in 1953, and found that, contrary to their expectations, the system would not reach thermodynamic equilibrium.

We study FPUT problem by applying the wave turbulence theory. We find that the resonant interactions of SIX waves does lead to irreversible energy mixing and eventually to the thermalization of the energy in the spectrum. We consider FPUT with quadratic (alpha FPUT model) and qubic (beta FPUT model) nonlinearities. We predict that for the alpha FPUT model the time scale to reach thermal equilibrium is of the order of 1/alpha^8. For the beta FPUT model the time to reach equipartitiion is of the order of 1/beta^4. This is why the emergence of equipartition requires such a long time, inaccessible in the fifties.

These results were obtained in collaboration with Miguel Onorato, Lara Vozella and Davide Proment

Becca Thomases (UC Davis)

Microorganism locomotion in viscoelastic fluids

Many important biological functions depend on microorganisms' ability to move in viscoelastic fluids such as mucus and wet soil. The effects of fluid elasticity on motility remain poorly understood, partly because, the swimmer strokes depend on the properties of the fluid medium, which obfuscates the mechanisms responsible for observed behavioral changes. In this study, we use experimental data on the gaits of the algal cell C. reinhardtii swimming in Newtonian and viscoelastic fluids as inputs to numerical simulations that decouple the swimmer gait and fluid type in order to isolate the effect of fluid elasticity on swimming. In viscoelastic fluids, cells employing the Newtonian gait swim faster but generate larger stresses and use more power, and as a result the viscoelastic gait is more efficient. Furthermore, we show that fundamental principles of swimming based on viscous fluid theory miss important flow dynamics: fluid elasticity provides an elastic memory effect which increases both the forward and backward speeds, and (unlike purely viscous fluids) larger fluid stress accumulates around flagella moving tangent to the swimming direction, compared to the normal direction.

Charles Doering (U. Michigan)

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

For quantities of interest in a dynamical system governed by differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization. The problems of finding maximal trajectories and minimal auxiliary functions are in fact strongly dual so auxiliary functions can produce arbitrarily sharp upper bounds on maximal time averages. They also define volumes in phase space where maximal trajectories must lie. For polynomial equations of motion auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz and Kuramoto Sivashinsy equations. This is joint work with Ian Tobasco and David Goluskin.

Minh Binh Tran (UW)

Some recent progress on wave turbulence and quantum kinetics

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales. Despite the fact that wave fields in nature are enormously diverse, there is a common mathematical concept that can be used to describe the processes of random wave interactions: the wave kinetic equations. After the production of the first Bose-Einstein Condensates (BECs), there has been an explosion of physics research on the kinetic theory associated to BECs and their thermal clouds. In this talk, we will summarize our recent progress on this topic.

Aaron Fogelson (Utah)

Continuum models of intravascular blood clot formation in arteries

Blood clots which form in the major arteries supplying the heart and brain with blood are the cause of most heart attacks and many strokes. These clots are made up largely of aggregates of blood platelets that are adherent to the vessel wall. We present several continuum models of this process. All of the models involve interactions among a viscous, incompressible fluid; populations of non-activated and activated platelets; activating chemicals; and the vessel walls. Adhesion of platelets to the injured wall and cohesion between activated platelets is modeled using distributions of elastic links which generate stresses that can strongly influence the fluid motion. Platelets are much smaller than the vessels in which the clots are developing, but events on the platelet scale have dramatic effects on those at the vessel scale. We first present a two-scale, single phase model in which both the platelet microscale and the vessel macroscale are treated and in which the fluid and platelets all move in the same velocity field. We show how the behavior of the two-scale model can be captured in a macroscale-only model, and we show examples of clot development as predicted by that model. Despite capturing important aspects of clot development, that model has significant limitations which can be traced to the assumption that there is no relative motion between the fluid and the platelets making up the aggregate. We extend that model to a multi-phase mixture model in which bound platelets move in a different velocity field than do the fluid and individual platelets, and show that it overcomes the limitations of the single phase model.