In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.
In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudospectral algorithms were developed for inverting the LaplaceBeltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, wellconditioned solvers for the Maxwell equations will rely on these algorithms.
Fast advection asymptotics for a stochastic reactiondiffusionadvection equation in a twodimensional bounded domain will be discussed. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.
We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phasefield model, consists of the AllenCahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime.
The second model type is a nonlinear free boundary problem for a KellerSegel type system of PDEs in 2D with area preservation and curvature entering the boundary conditions. We find an analytic oneparameter family of radially symmetric standing wave solutions (corresponding to a resting cell) as solutions to a Liouville type equation. Using topological tools, traveling wave solutions (describing steady motion) with noncircular shape are shown to bifurcate from the standing waves at a critical value of the parameter. Our bifurcation analysis explains, how varying a single (physical) parameter allows the cell to switch from rest to motion.
The work was done jointly with J. Fuhrmann, M. Potomkin, and V. Rybalko.
We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L^1 norm of the velocity field. Existing results in the literature use an L^p norm with p>1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.
I shall report recent results concerning the nonrelativistic limit of VlasovMaxwell systems in a large time. The talk is based on joint works with D. HanKwan and F. Rousset.
Coupled oscillators arise in contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London. Generally, such systems are either studied for a small number of oscillators or in the infinite oscillator, mean field limit. The dynamics of large but finite networks of oscillators is largely unknown. Kinetic theory was developed by Boltzmann and Maxwell to show how microscopic Hamiltonian dynamics of particles could account for the thermodynamic properties of gases. Here, I will show how concepts of kinetic theory and statistical field theory can be applied to deterministic coupled oscillator and neural systems to compute dynamical finite system size effects.
The leastaction problem for geodesic distance on the `manifold' of fluidblob shapes exhibits instability due to microdroplet formation. This reflects a striking connection between Arnold's leastaction principle for incompressible Euler flows and geodesic paths for Wasserstein distance. A connection with fluid mixture models via a variant of Brenier's relaxed leastaction principle for generalized Euler flows will be outlined also. This is joint work with JianGuo Liu and Dejan Slepcev.
Data assimilation is a method to estimate and predict evolution of physical system by integrating computational models and observations sampled from the evolving system. Prediction can be extremely challenging when the underlying physical system is highly nonlinear, corresponding model is highdimensional and complex, and observations are not only nonlinear but also heterogeneous and inhomogeneous. In the context of data assimilation, impact of selected sets of observations on the optimal estimation can be quantified by information theory. The concept can be extended to evaluate the impact on forecast skill. In other words, forecast skill can be traced back to the observations used in the state estimate using an adjoint technique that can be either explicit or ensemble based. Forecast Sensitivity to Observations (FSO) is a diagnostic tool that complements traditional data denial of the observing system experiments. In this talk, we will present the FSO in the operational numerical weather prediction. We will also discuss the effect of observation and model biases on FSO.
We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses. That is, the medium becomes effectively homogeneous in the largescale limit of the dynamics of solutions to the PDE. Hypotheses that guarantee this include fairly general stationary ergodic KPP reactions, as well as homogeneous ignition reactions in up to three dimensions perturbed by radially symmetric impurities distributed according to a Poisson point process.
In contrast to the original (secondorder) reactiondiffusion equations, the limiting "homogenized" PDE for this model are (firstorder) HamiltonJacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense. A key ingredient is a novel relationship between spreading speeds and front speeds for these models (as well as a proof of existence of these speeds), which can be thought of as the inverse of a wellknown formula in the case of periodic media, but we are able to establish it even for more general stationary ergodic media.
In a variety of applied settings, understanding the mechanisms of polymerization remains an open and important problem. In the composites industry, for example, different processing conditions change the microstructure of polymeric materials, which can ultimately impact the final material properties. Historically, however, a detailed understanding of this microstructure has been difficult to achieve because its inherent randomness is modeled in terms of highdimensional atomistic models.
To address this problem, I discuss a coarsegraining approach that transforms a discrete representation of polymerization models into a lowerdimensional and continuous framework. The main idea is to reformulate the polymerization process in terms of a discrete master equation whose generic structure can represent the different physical processes that dominate crosslinking. I show that appropriate limits of this master equation yield a system of generic, firstorder PDEs for the probability of different substructures within the polymerized network. Using the method of characteristics, I then discuss how this model can be transformed into a system of nonlinear ODEs, which in some cases can be solved exactly. In a related vein, I discuss how this approach provides a deeper understanding of analytical models first posed by Flory and show numerical results that confirm the accuracy of the approach in limiting cases.
Subriemannian geometry has its roots in optimal control problems. The CaratheodoryChowRashevskii theorem on accessibility also places the subject in contact with an axiomatic approach to macroscopic thermodynamics. Explicit integrability of optimal control problems in this context is of interest. As in the case for integrability questions in mechanics, here too symmetries and conservation laws have a key role. In this talk we discuss model problems and results pertaining to such questions in isolated systems and ensembles of interacting systems. Of special interest is the stochastic control problem of determining thermodynamic cycles that draw useful work from fluctuations. This work is in collaboration with PhD student Yunlong Huang, and Dr. Eric Justh of the Naval Research Laboratory.
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