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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