Geometric aspects of integrable systems and optimal control
, Mathematics, University of Michigan
In this talk I will discuss the dynamics and geometry of certain integrable Hamiltonian systems and their relationship to optimal control and the maximum principle. I will discuss in particular the Toda lattice equations, the generalized rigid body equations and their symmetric representation and flows on Stiefel manifolds. In addition I will discuss a class of integrable flows on the symmetric matrices which are distinct from the Toda flows.
Degenerate relative equilibria and the concept of criticality in fluid mechanics
, University of Surrey
A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria.
In this talk the nonlinear problem near a degenerate relative equilibrium is considered. The degeneneracy creates a saddle-centre and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. These properties are illuminated by using the Thom-Boardman theory of singularities for smooth mappings. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension with an -dimensional abelian symmetry group.
Surprisingly, this theory turns out to model in a precise way the concept of criticality in shallow water hydrodynamics. The classical theory will be recalled and it will be shown how it can be generalized to a wide range of fluid flows. For example, the theory leads to the existence of a pervasive new family of steady dark solitary waves in shallow water.
TJB (2004) Superharmonic instability, homoclinic torus bifurcation and water-wave breaking, J. Fluid Mech. 505: 153--162.
TJB & N.M. Donaldson (2005) Degenerate periodic orbits and homoclinic torus bifurcation, Phys. Rev. Lett. 95: 104301.
TJB & N.M. Donaldson (2006) Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, J. Fluid Mech. 565: 381--417.
TJB (2006) Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation, Preprint.
Strongly nonlinear wave models
, University of North Carolina
Some recently developed model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids will be discussed. These models follow from the original Euler theory under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. Removing the traditional weak nonlinearity balance with dispersion is necessary to improve agreement with the Euler dynamics of large amplitude waves, which advances in experimental detection show to occur commonly in geophysical situations. The new models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of Korteweg--de Vries (KdV) or Intermediate Long Wave (ILW) type, respectively for shallow and deep water configurations. Further asymptotic approximations that strive to maintain predominance of nonlinear effects result in models that can be completely integrable. The implication of this theoretical feature for numerical schemes as well as its analogies with point vortex methods and singularity formation in vortex patches will be illustrated.
Weak solutions of hydrodynamic equations
I will discuss weak solutions for the Euler equations and the QG
equations and the Onsager conjecture for anomalous dissipation of energy.
EPDiff and Computational Anatomy
, Imperial College
The Euler-Poincare Equation for Diffeomorphisms (EPDiff), which
specialises to the N-dimensional Camassa-Holm equation when the energy
functional is the H_1 norm for velocity, arises very naturally out of the problem of optimally controlling shapes using diffeomorphic flows. This has important applications in computational anatomy as it provides a way
of defining the distance between anatomical shapes. I shall give a
geometric overview of the connection with EPDiff, and show some new numerical results applied to matching curves and surfaces using a particle-mesh discretisation.
Twist & Shout: maximal enstrophy generation in the 3D Navier-Stokes equations
It is still not known whether solutions to the 3D Navier-Stokes equations for incompressible flows in a finite periodic box can become singular in finite time. (Indeed, this question is the subject of one of the $1M Clay Prize problems.) It is known that a solution remains smooth as long as the enstrophy, i.e., the mean-square vorticity, of the solution is finite. The generation rate of enstrophy is given by a functional that can be bounded using elementary functional estimates. Those estimates establish short-time regularity but do not rule out finite-time singularities in the solutions. In this work we formulate and solve the variational problem for the maximal growth rate of enstrophy and display flows that generate enstrophy at the greatest possible rate. Implications for questions of regularity or singularity in solutions of the 3D Navier-Stokes equations are discussed. This is joint work with Lu Lu (Wachovia Investments).
Lagrangian Coherent Structures in Ocean and Atmospheric Dynamics
In aperiodic geophysical flows, particle motion may appear to be chaotic and unstructured. However, using finite time Lyapunov exponents we are able to detect sharp separatrices that determine the structure of the flow. These separatrices elucidate the mechanisms by which particle transport and mixing occur. Computing these structures for oceanic and atmospheric flows allows us to visualize the underlying mixing processes by revealing many of the familiar elements from classical geometric dynamics including hyperbolic points, intersections of stable and unstable manifolds, homoclinic tangles, and transport via lobe dynamics. In particular, we see that lobe dynamics is a dominant mechanism for transport in hurricanes.
The d-bar Formalism and Applications to Imaging, Moving Boundaries and Integrability
A new method for inverting analyticaly certain integrals will be reviewed. This method involves the spectral analysis of an appropriate linear eigenvalue problem and the formulation of a local d-bar problem. Aplications include imaging and the solution of initial boundary value problems for PDEs involving a moving boundary. In addition, the long standing open problem of constructing integrable multidimensional nonlinear PDEs will be discussed. It will be shown that the Cauchy problem for such PDEs leads to a nonlocal d-bar problem. Integrable nonlinear PDEs in 3+1 compatible with the Laplace equation will be presented.
Regularization modeling of turbulent mixing; sweeping the scales
Mathematical regularization of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modeled system of equations. We will consider the large eddy interpretation of two such mathematical regularization principles, i.e., Leray and LANS-alpha regularization. The Leray principle introduces a smoothed transport velocity as part of the regularized convective nonlinearity. The LANS-alpha principle extends the Leray formulation in a natural way in which a filtered Kelvin circulation theorem, incorporating the smoothed transport velocity, is explicitly satisfied. These regularization principles imply a modeling of turbulent transport in which small-scale flow-features are primarily swept by the larger scales.
The implied subgrid closures were implemented in large eddy simulations of turbulent mixing in a plane mixing layer. Comparison with filtered direct numerical simulation data and with predictions obtained from popular dynamic eddy-viscosity modeling shows that the mathematical regularization models provide more accuracy at a lower computational cost. In particular, the regularization models perform especially well in capturing the flow features characteristic of the smaller resolved scales. A level-set analysis is applied to quantify the scalar mixing. Effects of rotation and stratification on the efficiency of turbulent mixing were investigated.
Ortho-normal quaternion frames and Lagrangian evolution equations
More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. This talk will survey some of the ideas in this area that Darryl and I have worked on together this past two years. It will be shown that quaternions provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. If time permits, it will be shown how these ideas can be applied to the three-dimensional Euler fluid equations and how they impact on the issue of finite-time blow-up.
Multi-dimensional Compactons in Nonlinear Wave Equations
Collaborator: Philip Rosenau
(School of Mathematical Sciences, Tel-Aviv University)
Most solitions are one-dimensional waves, even in mul-
tidimensional equations. Solitons are nonlinear traveling
waves where the nonlinearity and dispersion are balanced
to create a stable coherent local environment so that that
the solitions maintain their coherence when colliding with
other solitions. Because the same dispersive operators
are much stronger to two and three dimensions than in
one dimension, this balance is usually lost when a one-
dimensional equation is generalized to higher dimensions.
We restore the balance by making the dispersion weaker
in a class of KdV and regularized long wave equations to
create fully two and three dimensional solutions.
Mathematical models of credit correlations
, Merrill Lynch International, London
In this talk we introduce the financially important and mathematically challenging problem of credit correlation.
After a brief historical overview, we describe dynamic and static factor models for solving this problem and discuss their pros and cons. We show how these models can be calibrated to the market and used for the pricing of standard and bespoke tranches.
The Stress-Energy-Momentum Tensor
, Control and Dynamical Systems, California Institute of Technology
Collaborators: Marco Castrillon Lopez
(Mathematics, UCM, Madrid) and Mark Gotay
(Mathematics, University of Hawaii)
The stress-energy-momentum (SEM) tensor is one of the most basic objects in classical field theory, yet it is one of the most abused and misunderstood. We begin by reviewing the Gotay-Marsden multisymplectic approach to the classical Belinfante-Rosenfeld-Hilbert theory that relates the classical canonical SEM tensor to its formula as the derivative of the Lagrangian with respect to the metric. The relation with the multimomentum map that relates the SEM tensor to multimomentum fluxes is also treated. Interestingly, virtually any field theory can be made diffeomorphism invariant by using the Kuchar trick to introduce new diffeomorphisms as fields. Their conjugate momenta then turn out to be exactly the SEM tensor and also reveal a nice generalization of the relation between the Cauchy and Piola-Kirchhoff stress tensor as well as the Doyle-Ericksen formula in nonlinear elasticity.
Lagrangian Coherent Structures
Lagrangian Coherent Structures are invariant material manifolds that
can be identified through careful study of particle mixing behavior.
They are also defined as ridges of finite time Lyapunov exponent
(FTLE) fields. In this talk I will present results for LCS
computations that my student B. Cardwell has performed for a two
dimensional flow around an airfoil at moderately low Reynolds
numbers. Various stable and unstable manifolds are identified and
their implications in global flow features are investigated.
Fluctuation Spectra of Plasmas and Fluids
The spectrum of electron phase space density fluctuations of a plasma is
calculated by a novel method that parallels conventional calculations of the
partition function in statistical physics. Expressions for the electric field
fluctuations and the closely related form factor agree with existing
results for thermal plasmas. The method clears up ambiguities about
equipartition and provides a new expression for the spectrum of
electrostatic phase space density fluctuations about stable
non-Maxwellian equilibria. The method is of general utility and
applies to a large class of Hamiltonian systems. It has been applied
to vortex dynamics -- time permitting, these results will be described.
Infinite dimensional symmetry groups
In this talk, I will present new, constructive methods, based on moving frames, for determining the structure of infinite-dimensional (and finite-dimensional) symmetry groups of differential equations, along with their associated algebras of differential invariants. Time permitting, some initial applications to the construction of exact solutions and the design of symmetry-preserving numerical integrators will also be discussed.
The LANS-alpha turbulence model in primitive-equation ocean modeling
Collaborators: Matthew Hecht, Darryl Holm, and Beth Wingate
POP, the Parallel Ocean Program developed and maintained by Los Alamos
National Laboratory, is widely used by the ocean and climate modeling
community. Like all numerical models, computational time limits the
spatial resolution at which POP can operate; standard simulations use
grids of 0.5 to 1 degree in latitude and longitude. This resolution does
not capture the motion of eddies at the Rossby radius of deformation, and
thus lacks the correct energy cascade and heat transport at these scales.
Simulations using the Lagrangian-averaged Navier Stokes-alpha (LANS-alpha)
turbulence parameterization in the POP ocean model resemble higher
resolution simulations of standard POP in statistics like kinetic energy,
eddy kinetic energy, and potential temperature fields. The LANS-alpha
model accomplishes this improvement through an additional nonlinear term
and a smoothed advecting velocity.
I will discuss my implementation of LANS-alpha in the POP ocean model,
and results using an idealized channel domain that invokes the
baroclinic instability. I also show comparisons between the alpha-model
and the Gent-McWilliams (GM) model, a model which arguably was one of the
most important developments in ocean modeling in the last decade. The
alpha model is shown to make superior predictions of eddy kinetic energy
equivalent to that produced at twice the resolution with no model, while
the GM model excels at description of the tracers but suppresses eddy
On the qualitative behavior of the solutions to the 2-D Navier-Stokes equation
The planar Navier-Stokes equation exhibits, in absence of external forces, a trivial asymptotics in time. Nevertheless the appearence of stable coherent structures suggests a non-trivial intermediate asymptotics which should be explained in terms of the equation itself.
In the present talk I present and study a new class of equations, based on a projection procedure, motivated by a separation of different time scales and the gradient structure of the Navier-Stokes equation. The solutions of such equations have asymptotic states which are invariant for the Euler flow and solve a nonlinear elliptic problem.
The results I am presenting are due to a collaboration of the author with E. Caglioti and F. Rousset.
Mathematical models for self-aggregation of particles:
from nano- to millimeter scales (or cheerios at 50 nanometers)
(Mathematics, Colorado State University)
Collaborator: Darryl Holm
(Mathematics, Imperial College)
We derive an evolution equations for self-assembly of particles at the nano-scale. First, we present results of numerical simulations showing that energy density of a realistic particle clump of an arbitrary shape is described by a surprisingly simple formula, and that the continuum approximation may be used for clumps with as few as five particles. Next, we deduce an evolution equation for density evolution for in the case of central potential and describe analytically stationary and spatio-temporal solutions forming the base of the dynamics. Finally, we suggest an evolution equations for particles with interaction potential dependent not only on their density, but also on the mutual orientation. In the process, we also derive the analogue of Darcy's law and corresponding equation of motion for an arbitrary geometric quantity.
The Euler-Weil-Petersson equations
This is a progress report on joint work with
François Gay-Balmaz and Jerrold Marsden. It presents
the basic approach to the universal Teichmüller
space and discusses the problems this point of view
produces when the Weil--Petersson metric is introduced.
Then it discusses a new approach due to Takhtajan and
Teo that solves the convergence problem of the formula
for the Weil-Petersson metric. In this point of view, the
universal Teichmüller space is a strong
Köhler--Einstein Hilbert manifold with negative constant
Ricci curvature and negative sectional and holomorphic
sectional curvatures. The geodesic spray of this metric
is discussed and studied. The relation of the universal
Teichmüller space to the Bott-Virasoro group is also
Alpha Sub-grid Scale Models of Turbulence and Inviscid Regularization
, Weizmann Institute of Science and University of California, Irvine
In recent years many analytical sub-grid scale models of turbulence
were introduced based on the Navier--Stokes-alpha model (also known
as a viscous Camassa--Holm equations or the Lagrangian Averaged
Navier--Stokes-alpha (LANS-alpha)). Some of these are the
Leray-alpha, the modified Leray-alpha, the simplified Bardina-alpha
and the Clark-alpha models. In this talk we will show the global
well-posedness of these models and provide estimates for the
dimension of their global attractors, and relate these estimates to
the relevant physical parameters. Furthermore, we will show that up
to certain wave number in the inertial range the energy power
spectra of these models obey the Kolmogorov -5/3 power law, however,
for the rest the inertial range the energy spectra are much steeper.
This observation makes these alpha models more computable than the
Navier--Stokes equations and consequently are adequate sub-grid
scale models of turbulence.
In addition, we will show that by using these alpha models as
closure models to the Reynolds averaged equations of the
Navier--Stokes one gets very good agreement with empirical and
numerical data of turbulent flows in infinite pipes and channels.
We also observe that, unlike the three-dimensional Euler equations
and other inviscid alpha models, the inviscid simplified Bardina
model has global regular solutions for all initial data. Inspired by
this observation we introduce new inviscid regularizing schemes for
the three-dimensional Euler and Navier--Stokes equations, which does
not require, in the latter case, any additional boundary condition.
This same kind of inviscid regularization is also used to regularize
the Surface Quasi-Geostrophic model.
Groupoid symmetry for Einstein's equations?
The solutions of the constraint equations in the 3+1 formulation of
Einstein's equations appears to be the zero set of a momentum map, but
this does not quite work when the symmetry group is taken to be the
4-dimensional diffeomorphisms. In this talk, I will report on ongoing
work with Christian Blohmann (Berkeley) and Marco Cezar Fernandes
(Brasilia). We are attempting to show that the constraint set is the
zero set of the momentum map for the action of a groupOID, such as the
groupoid of diffeomorphisms between all pairs of hypersurfaces in
The LANS-alpha model of ocean dynamics: fluid and numerical stability
Collaborators: Matthew Hecht, Darryl Holm, and Mark Petersen
In this talk I give an overview of efforts over the past 5 years to
understand LANS-alpha as a model of ocean dynamics. In today's global
ocean models, models that are used as the ocean component in
IPCC-class climate models, the most important length scale is the
Rossby deformation radius. The highest resolutions achieved today are
globally 1/10th degree, a resolution that well-resolves the
deformation radius. But when these models are run in climate
simulations the grid is much coarser.
I will discuss the fluid stability problem associated with capturing order
one deformation radius dynamics, the numerical stability issues we
confronted while making this viable for an ocean model, and give
preliminary results of the alpha model for rotating and/or stratified
flow. I will also discuss some outstanding questions related to
Metamorphoses for Pattern Matching
Collaborators: D. Holm and A. Trouvé.
Pattern matching is the search for optimal correspondences between to shapes, images or more general deformable structures. It has been successfully addressed, in a variety of applications, by formulating it as a variational problem in groups of diffeomorphisms.
We discuss here a specific approach to this problem which is especially important when the objects of interest do not form a single orbit under diffeomorphic action, which is typical when, for example, images are compared. Instead of having recourse to inexact matching methods, where one looks for an optimal approximation of a target object by a deformation of a template one, the idea is to optimize an evolution that relates the two objects, in which diffeomorphic transformations are coupled with intrinsic variations in the object space. We will formulate a general setting for this framework, called Metamorphosis, within which we will derive general Euler-Lagrange equations. We will also discuss, in several particular cases of interest, the existence of solutions of these equations, as well as the existence of solutions of the corresponding variational problem.