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ACMS Abstracts: Fall 2015

Li Wang (UCLA)

Singular shocks: From particle-laden flow to human crowd dynamics

In this talk, we will present two examples in which singular shock arises. The first example, gravity-driven thin film flow with a suspension of particles down an incline, is described by a system of conservation laws equipped with an equilibrium theory for particle settling and resuspension. Singular shock appears in the high particle concentration case that relates to the particle-rich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure, and extend to the finite volume case, which leads to a linear relationship between the shock front with time to the one-third power. The second example, a panicking crowd with a spread of fear, is modeled via ``emotional contagion”. Singular shock happens in an extreme case whose continuum limit is a pressure less Euler equation. Such system is then modified with a nonlocal alignment to regularize the singularity. We will discuss the hierarchy of models and their mathematical properties. Novel numerical methods will be presented for both examples.

Wai Tong (Louis) Fan

Reflected diffusions with partial annihilations on a membrane

Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce a class of interacting particle systems that can model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales, we obtain the hydrodynamic limits and the fluctuation limits for these systems. Proving these two types of limits represents establishing the law of large numbers and the central limit theorem, respectively, for the time-trajectory of the particle densities. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is joint work with Zhen-Qing Chen. This talk will focus on modeling methods and applications. A second talk on the probabilistic techniques involved in the proofs will be given in the Probability Seminar on Oct 15.

Wenjia Jing (Chicago)

Limiting distributions of random fluctuations in stochastic homogenization

In this talk, I will present some results on the study of limiting distributions of the random fluctuations in stochastic homogenization. I will discuss first a framework of such studies for linear equations with random potential. The scaling factor and the scaling limit of the homogenization error turn out to depend on the singularity of the Green’s function and the correlation structure of the random potential. I will also present some results that extend the scope of the framework to the setting of oscillatory differential operators and to some nonlinear equations. Such results find applications, for example, in uncertainty quantification and Bayesian inverse problems.

Arthur Evans (UW)

Ancient art and modern mechanics: using origami design to create new materials

The Japanese art of origami has been a purely aesthetic endeavor for hundreds of years, but recent decades have seen the application of cutting, creasing and folding to fields as diverse as architecture and nano-engineering. The key link between the artistry of paper-folding and the physics of cells and shells lies in the connection between geometry and mechanics. In this talk I will discuss the emergence of origami design for understanding the mechanics of thin structures, highlighting the physical and mathematical principles that drive the folding of a thin sheet. While much of origami-based engineering has relied on heuristic development, I will present here a method for generalizing material design in tessellated structures, and discuss some of the first steps in building a theory that adapts origami mechanics to non-Euclidean surfaces.

Alfredo Wetzel (UW)

Direct scattering for the Benjamin-Ono equation with rational initial data

The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable system with a known inverse scattering transform and can be viewed as a prototypical problem for the study of multi-dimensional integrable systems or Riemann-Hilbert problems with a nonlocal jump condition. In this talk, we propose a construction procedure for the scattering data of the BO equation for arbitrary rational initial conditions with simple poles, under mild restrictions. For this class of initial conditions, we are able to obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient. Lastly, we show that this procedure validates well-known formal results in the zero-dispersion limit.

Mohammed Lemou (Universite Rennes I)

A class of numerical schemes for multiscale parabolic problems

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as the oscillation parameter goes to 0, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, known such homogenization schemes are in general not accurate for both the highly oscillatory regime and the non oscillatory regime. In this paper, we introduce an Asymptotic Preserving method based on an exact micro-macro decomposition of the solution which remains consistent for both regimes.

Victor Zavala (UW)

Large-scale nonlinear programming and applications to energy networks

We present advances in nonlinear programming that enable the solution of large-scale problems arising in the control of electrical and natural gas networks. Our advances involve new strategies to deal with negative curvature and Jacobian rank deficiencies in a matrix-free setting and the development of scalable numerical linear algebra strategies capable of exploiting embedded problem structures.

Melvin Leok (UCSD)

Geometric numerical integration and computational geometric mechanics

Symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes.

In this talk, we will introduce a systematic method of constructing geometric integrators based on a discrete Hamilton's variational principle. This involves the construction of discrete Lagrangians that approximate Jacobi's solution to the Hamilton-Jacobi equation. Jacobi's solution can be characterized either in terms of a boundary-value problem or variationally, and these lead to shooting-based variational integrators and Galerkin variational integrators, respectively. We prove that the resulting variational integrator is order-optimal, and when spectral basis elements are used in the Galerkin formulation, one obtains geometrically convergent variational integrators.

We will also introduce the notion of a boundary Lagrangian, which is analogue of Jacobi's solution in the setting of Lagrangian PDEs. This provides the basis for developing a theory of variational error analysis for multisymplectic discretizations of Lagrangian PDEs. Equivariant approximation spaces will play an important role in the construction of geometric integrators that exhibit multimomentum conservation properties, and we will describe two approaches based on spacetime generalizations of Finite-Element Exterior Calculus, and Geodesic Finite Elements on the space of Lorentzian metrics.

Qin Li (UW-Madison)

Kinetic-fluid coupling: transition from the Boltzmann to the Euler

Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime co-exist, how to couple the two systems remains an open problem. The key is to understand the half-space problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the well-posedness of a class of half-space equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.

Matthias Schlottbom (University of Münster)

On Galerkin schemes for time-dependent radiative transfer

The numerical solution of time dependent radiative transfer problems is challenging, both, due to the high dimension and the anisotropic structure of the underlying integro-partial differential equation. Starting from an appropriate variational formulations, we propose a general strategy for designing numerical methods based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to inherit basic properties like the conservation of mass and exponential stability from the continuous level. We also present the basic approximation error estimates. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure as more standard first order hyperbolic systems in acoustics or electrodynamics. This allows us to generalize the main arguments of the numerical analysis of such applications to the radiative transfer problems under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle and a finite element discretization in space. This is a joint work with Herbert Egger (TU Darmstadt)

Theodoros Katasounis (KAUST)

A posteriori error control and adaptivity for Schrodinger equations

We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrodinger-type equations. For the discretization in time we use the Crank-Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrodinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrodinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrodinger equations. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant. The analysis is extended also for the nonlinear Schrodinger eq.