Publications
335 results found
Holm DD, Tyranowski TM, 2016, Variational Principles for Stochastic Soliton Dynamics, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol: 472, ISSN: 0080-4630
We develop a variational method of deriving stochastic partial differential equations whose solutions followthe flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singularsolutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method.These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equationpersist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticityto finite dimensional solutions of stochastic partial differential equations (SPDE). In particular, some choices ofstochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations,or CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, althoughthis behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincar´estructure of the CH equation (parametric stochastic deformations, or P-SD), and it also does not occur forpeakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the scienceof stochastic deformations of finite-dimensional approximations of evolutionary PDE and the sensitivity of theresulting solutions to the choices made in stochastic modelling.
Holm DD, 2015, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 471, ISSN: 1364-5021
This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler-Boussinesq and quasi-geostropic approximations.
Bruveris M, Holm DD, 2015, Geometry of image registration: The diffeomorphism group and momentum maps, Geometry, Mechanics, and Dynamics, Editors: chang, Holm, Patrick, Ratiu, Publisher: Springer New York, Pages: 19-56, ISBN: 9781493924400
These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.
Gay-Balmaz F, Holm DD, 2014, A geometric theory of selective decay with applications in MHD, NONLINEARITY, Vol: 27, Pages: 1747-1777, ISSN: 0951-7715
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- Citations: 12
Holm DD, Ivanov RI, 2014, Matrix G-strands, NONLINEARITY, Vol: 27, Pages: 1445-1469, ISSN: 0951-7715
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- Citations: 4
Gay-Balmaz F, Holm DD, Ratiu TS, 2014, Integrable <i>G</i>-strands on semisimple Lie groups, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 47, ISSN: 1751-8113
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- Citations: 1
Holm DD, Ivanov RI, 2014, Euler-Poincaré equations for G-Strands, Conference on Physics and Mathematics of Nonlinear Phenomena (PMNP), Publisher: Institute of Physics (IoP), ISSN: 1742-6588
The G-strand equations for a map Bbb R × Bbb R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t, s) : Bbb R × Bbb R → G, where t and s are the independent variables of the G-strand equations. The Euler-Poincaré reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra and its co-algebra, * with respect to the pairing provided by the variational derivatives of the Lagrangian.We review examples of different G-strand constructions, including matrix Lie groups and diffeomorphism group. In some cases the G-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.
Cotter CJ, Holm DD, Jacobs HJ, et al., 2014, A jetlet hierarchy for ideal fluid dynamics, J Phys A, Vol: 47
Cotter CJ, Holm DD, 2014, Variational formulations of sound-proof models, Quarterly Journal of the Royal Meteorological Society
Burnett CL, Holm DD, Meier DM, 2013, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 469, ISSN: 1364-5021
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- Citations: 10
Gibbon JD, Holm DD, 2013, Stretching and folding diagnostics in solutions of the three-dimensional Euler and Navier-Stokes equations, Procedia IUTAM, Vol: 9, Pages: 25-31, ISSN: 2210-9838
Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the context of the vector B = ∇q × ∇θ where, in atmospheric physics, θ is a temperature, q = ω · ∇θ is the potential vorticity, and ω = curl u is the vorticity. These ideas are then discussed in the context of the full compressible Navier-Stokes equations where q is taken in the form q = ω · ∇ f (ρ). In the two cases f = ρ and f = ln ρ, q is shown to satisfy the quasi-conservative relation ∂t q + div J = 0.
Gibbon JD, Holm DD, 2013, Stretching and folding processes in the 3D Euler and Navier-stokes equations, Procedia IUTAM, Vol: 9, Pages: 25-31, ISSN: 2210-9838
Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the contextof the vector B = ∇q×∇θ where, in atmospheric physics, θ is a temperature, q = ω ·∇θ is the potential vorticity, and ω = curluis the vorticity. These ideas are then discussed in the context of the full compressible Navier-Stokes equations where q is taken inthe form q = ω ·∇ f(ρ). In the two cases f = ρ and f = lnρ, q is shown to satisfy the quasi-conservative relation ∂tq+divJ = 0
Holm DD, Noakes L, Vankerschaver J, 2013, Relative geodesics in the special Euclidean group, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 469, ISSN: 1364-5021
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- Citations: 4
Cotter CJ, Holm DD, 2013, A variational formulation of vertical slice models, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 469, ISSN: 1364-5021
A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin–Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler–Boussinesq equations with a constant temperature gradient in the y-direction (the Eady–Boussinesq model), which is an idealized problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model. Unlike the incompressible model, the compressible model does not produce solutions that are also solutions of the three-dimensional equations, but it does reduce to the Eady–Boussinesq model in the low Mach number limit. Hence, the new model could be used in asymptotic limit error testing for compressible weather models running in a vertical slice configuration.
Holm DD, Putkaradze V, Tronci C, 2013, COLLISIONLESS KINETIC THEORY OF ROLLING MOLECULES, KINETIC AND RELATED MODELS, Vol: 6, Pages: 429-458, ISSN: 1937-5093
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- Citations: 1
Gay-Balmaz F, Holm DD, Ratiu TS, 2013, GEOMETRIC DYNAMICS OF OPTIMIZATION, COMMUNICATIONS IN MATHEMATICAL SCIENCES, Vol: 11, Pages: 163-231, ISSN: 1539-6746
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- Citations: 11
Gibbon JD, Holm DD, 2013, Enstrophy bounds and the range of space-time scales in the hydrostatic primitive equations, Physical Review E, Vol: 87, ISSN: 1539-3755
The hydrostatic primitive equations (HPEs) form the basis of most numerical weather, climate, and global ocean circulation models. Analytical (not statistical) methods are used to find a scaling proportional to (NuRaRe)1/4 for the range of horizontal spatial sizes in HPE solutions, which is much broader than is currently achievable computationally. The range of scales for the HPE is determined from an analytical bound on the time-averaged enstrophy of the horizontal circulation. This bound allows the formation of very small spatial scales, whose existence would excite unphysically large linear oscillation frequencies and gravity wave speeds.
Gay-Balmaz F, Holm DD, 2013, Selective decay by Casimir dissipation in inviscid fluids, NONLINEARITY, Vol: 26, Pages: 495-524, ISSN: 0951-7715
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- Citations: 19
Henry D, Holm D, Ivanov R, 2013, On the persistence properties of the cross-coupled Camassa-Holm system, Journal of Geometry and Symmetry in Physics, Vol: 32, Pages: 1-13, ISSN: 1312-5192
In this paper we examine the evolution of solutions, of a recently- derived system of cross-coupled Camassa-Holm equations, that initially have compact support. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For the solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.
Holm DD, Munn J, Stechmann SN, 2013, Singular Solutions of Euler-Poincaré Equations on Manifolds with Symmetry, Pages: 267-316, ISSN: 2194-1009
The Euler-Poincaré equation EPDiff governs geodesic flow on the diffeomorphisms with respect to a chosen metric, which is typically a Sobolev norm on the tangent space of vector fields. For a strong enough norm, EPDiff admits singular solutions, called "diffeons," whose momenta are supported on embedded subspaces of the ambient space. Diffeons are true solitons for some choices of the norm. The diffeon solution itself is a momentum map. Consequently, the diffeons evolve according to canonical Hamiltonian equations.This paper examines diffeon solutions on Einstein spaces that are "mostly" symmetric, i.e., whose quotient by a subgroup of the isometry group is one-dimensional. An example is the two-sphere, whose isometry group SO3 contains S 1. In this situation, the singular diffeons are supported on latitudes of the sphere. For this S 1 symmetry of the two-sphere, the canonical Hamiltonian dynamics for diffeons reduces from integral partial differential equations to a dynamical system of ordinary differential equations for their co-latitudes. Explicit examples are computed numerically for the motion and interaction of the Puckons on the sphere with respect to the H 1 norm. We analyze this case and several other two-dimensional examples. From consideration of these two-dimensional spaces, we outline the theory for reduction of diffeons on a general manifold possessing a metric equivalent to the warped product of the line with the bi-invariant metric of a Lie group. © Springer Basel 2013.
Holm DD, Ivanov RI, 2013, <i>G</i>-Strands and Peakon Collisions on Diff(R), SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, Vol: 9, ISSN: 1815-0659
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- Citations: 3
Holm DD, Tronci C, 2012, Multiscale turbulence models based on convected fluid microstructure, Journal of Mathematical Physics, Vol: 53, ISSN: 1089-7658
The Euler-Poincaré approach to complex fluids is used to derive multiscale equations for computationally modeling Euler flows as a basis for modeling turbulence. The model is based on a kinematic sweeping ansatz (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest two-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modeling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.
Gibbon JD, Holm DD, 2012, Quasiconservation laws for compressible three-dimensional Navier-Stokes flow, Physical Review E, Vol: 86, ISSN: 1539-3755
We formulate the quasi-Lagrangian fluid transport dynamics of mass density ρ and the projection q=ω⋅∇ρ of the vorticity ω onto the density gradient, as determined by the three-dimensional compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of q cannot cross a level set of ρ. That is, in this formulation, level sets of ρ (isopycnals) are impermeable to the transport of the projection q.
Brody DC, Holm DD, Meier DM, 2012, Quantum splines, Physical Review Letters, Vol: 109, ISSN: 0031-9007
A quantum spline is a smooth curve parametrized by time in the space of unitary transformations, whose associated orbit on the space of pure states traverses a designated set of quantum states at designated times, such that the trace norm of the time rate of change of the associated Hamiltonian is minimized. The solution to the quantum spline problem is obtained, and is applied in an example that illustrates quantum control of coherent states. An efficient numerical scheme for computing quantum splines is discussed and implemented in the examples.
Holm DD, Vizman C, 2012, DUAL PAIRS IN RESONANCES, JOURNAL OF GEOMETRIC MECHANICS, Vol: 4, Pages: 297-311, ISSN: 1941-4889
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- Citations: 5
Holm DD, Ivanov RI, Percival JR, 2012, <i>G</i>-Strands, JOURNAL OF NONLINEAR SCIENCE, Vol: 22, Pages: 517-551, ISSN: 0938-8974
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- Citations: 11
Gay-Balmaz F, Holm DD, Meier DM, et al., 2012, Invariant Higher-Order Variational Problems II, JOURNAL OF NONLINEAR SCIENCE, Vol: 22, Pages: 553-597, ISSN: 0938-8974
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- Citations: 26
Cotter CJ, Holm DD, 2012, On Noether's Theorem for the Euler-Poincaré equation on the diffeomorphism group with advected quantities, Foundations of Computational Mathematics, ISSN: 1615-3375
Gay-Balmaz F, Holm DD, Putkaradze V, et al., 2012, Exact geometric theory of dendronized polymer dynamics, ADVANCES IN APPLIED MATHEMATICS, Vol: 48, Pages: 535-574, ISSN: 0196-8858
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- Citations: 6
Holm DD, Tronci C, 2012, EULER-POINCARE FORMULATION OF HYBRID PLASMA MODELS, COMMUNICATIONS IN MATHEMATICAL SCIENCES, Vol: 10, Pages: 191-222, ISSN: 1539-6746
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- Citations: 19
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