1,721,062 research outputs found
Lie-Poisson methods for isospectral flows
Full text linkThe theory of isospectral flows comprises a large class of continuous
dynamical systems, particularly integrable systems and Lie--Poisson systems.
Their discretization is a classical problem in numerical analysis. Preserving
the spectra in the discrete flow requires the conservation of high order
polynomials, which is hard to come by. Existing methods achieving this are
complicated and usually fail to preserve the underlying Lie--Poisson structure.
Here we present a class of numerical methods of arbitrary order for Hamiltonian
and non-Hamiltonian isospectral flows, which preserve both the spectra and the
Lie--Poisson structure. The methods are surprisingly simple, and avoid the use
of constraints or exponential maps. Furthermore, due to preservation of the
Lie--Poisson structure, they exhibit near conservation of the Hamiltonian
function. As an illustration, we apply the methods to several classical
isospectral flows.Comment: 29 pages, 9 figure
Integrability of point-vortex dynamics via symplectic reduction: a survey
Full text linkPoint-vortex dynamics describe idealized, non-smooth solutions to the
incompressible Euler equations on 2-dimensional manifolds. Integrability
results for few point-vortices on various domains is a vivid topic, with many
results and techniques scattered in the literature. Here we give a unified
framework for proving integrability results for , , or
point-vortices (and also more general Hamiltonian systems), based on symplectic
reduction theory. The approach works on any 2-dimensional manifold; we
illustrate it on the sphere, the plane, the hyperbolic plane, and the flat
torus. A systematic study of integrability is prompted by advances in
2-dimensional turbulence, bridging the long-time behaviour of 2D Euler
equations with questions of point-vortex integrability. A gallery of solutions
is given in the appendix.Comment: 26 pages, 4 figures, accepted in Arnold Math.
An efficient geometric method for incompressible hydrodynamics on the sphere
Full text linkWe present an efficient and highly scalable geometric method for
two-dimensional ideal fluid dynamics on the sphere. The starting point is
Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from
exploiting a tridiagonal splitting of the discrete spherical Laplacian combined
with highly optimized, scalable numerical algorithms. For time-stepping, we
adopt a recently developed isospectral integrator able to preserve the
geometric structure of Euler's equations, in particular conservation of the
Casimir functions. To overcome previous computational bottlenecks, we formulate
the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue
problems, efficiently solved by well-established linear algebra libraries. The
same tridiagonal splitting allows for computation of the stream matrix,
involving the inverse Laplacian, for which we design an efficient parallel
implementation on distributed memory systems. The resulting overall
computational complexity is per time-step for spatial
degrees of freedom. The dominating computational cost is matrix-matrix
multiplication, carried out via the parallel library ScaLAPACK. Scaling tests
show approximately linear scaling up to around cores for the matrix size
with a computational time per time-step of about seconds. These
results allow for long-time simulations and the gathering of statistical
quantities while simultaneously conserving the Casimir functions. We illustrate
the developed algorithm for Euler's equations at the resolution
Casimir preserving spectrum of two-dimensional turbulence
Full text linkWe present predictions of the energy spectrum of forced two-dimensional turbulence obtained by employing a structure-preserving integrator. In particular, we construct a finite-mode approximation of the Navier-Stokes equations on the unit sphere, which, in the limit of vanishing viscosity, preserves the Lie-Poisson structure. As a result, integrated powers of vorticity are conserved in the inviscid limit. We obtain robust evidence for the existence of the double energy cascade, including the formation of the -3 scaling of the inertial range of the direct cascade. We show that this can be achieved at modest resolutions compared to those required by traditional numerical methods
Generalized Hunter–Saxton equations, optimal information transport, and factorization of diffeomorphisms
No full textWe study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher’s information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalizations of the μ-Hunter–Saxton equation, used to model liquid crystals under the influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray.</br></br>
The descending property of the metrics is used to obtain a novel factorization of diffeomorphisms. Analogous to the polar factorization in optimal mass transport, this factorization solves an optimal information transport problem. It can be seen as an infinite-dimensional version of QR factorization of matrices
Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
No full textThe space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher--Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow
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