Publications
335 results found
Gay-Balmaz F, Holm DD, 2020, PREDICTING UNCERTAINTY IN GEOMETRIC FLUID MECHANICS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S, Vol: 13, Pages: 1229-1242, ISSN: 1937-1632
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- Citations: 3
Holm DD, Naraigh LO, Tronci C, 2020, A geometric diffuse-interface method for droplet spreading, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 476, ISSN: 1364-5021
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- Citations: 1
Holm DD, 2019, Stochastic Closures for Wave-Current Interaction Dynamics, Publisher: SPRINGER
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- Citations: 9
Foskett MS, Holm DD, Tronci C, 2019, Geometry of nonadiabatic quantum hydrodynamics, Acta Applicandae Mathematicae, Vol: 162, Pages: 63-103, ISSN: 0167-8019
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ℏ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ℏ≠0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.
Cotter C, Crisan D, Holm DD, et al., 2019, A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation, Publisher: arXiv
In this work, we apply a particle filter with three additional procedures(model reduction, tempering and jittering) to a damped and forcedincompressible 2D Euler dynamics defined on a simply connected bounded domain.We show that using the combined algorithm, we are able to successfullyassimilate data from a reference system state (the ``truth") modelled by ahighly resolved numerical solution of the flow that has roughly $3.1\times10^6$degrees of freedom for $10$ eddy turnover times, using modest computationalhardware. The model reduction is performed through the introduction of a stochasticadvection by Lie transport (SALT) model as the signal on a coarser resolution.The SALT approach was introduced as a general theory using a geometricmechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows onthe numerical implementation for SALT presented by Cotter et al, SIAMMultiscale Model. Sim. (2019) for the flow in consideration. The modelreduction is substantial: The reduced SALT model has $4.9\times 10^4$ degreesof freedom. Forecast reliability and estimated asymptotic behaviour of the particlefilter are also presented.
Crisan D, Flandoli F, Holm DD, 2019, Solution properties of a 3D stochastic euler fluid equation, Journal of Nonlinear Science, Vol: 29, Pages: 813-870, ISSN: 0938-8974
We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.
Holm D, 2019, Stochastic evolution of augmented Born–Infeld equations, Journal of Nonlinear Science, Vol: 29, Pages: 115-138, ISSN: 0938-8974
This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.
Holm DD, 2019, Stochastic parametrization of the Richardson triple, Journal of Nonlinear Science, Vol: 29, Pages: 89-113, ISSN: 0938-8974
A Richardson triple is an ideal fluid flow map (Formula presented.) composed of three smooth maps with separated time scales: slow, intermediate and fast, corresponding to the big, little and lesser whorls in Richardson’s well-known metaphor for turbulence. Under homogenization, as (Formula presented.), the composition (Formula presented.) of the fast flow and the intermediate flow is known to be describable as a single stochastic flow (Formula presented.). The interaction of the homogenized stochastic flow (Formula presented.) with the slow flow of the big whorl is obtained by going into its non-inertial moving reference frame, via the composition of maps (Formula presented.). This procedure parameterizes the interactions of the three flow components of the Richardson triple as a single stochastic fluid flow in a moving reference frame. The Kelvin circulation theorem for the stochastic dynamics of the Richardson triple reveals the interactions among its three components. Namely, (1) the velocity in the circulation integrand is kinematically swept by the large scales and (2) the velocity of the material circulation loop acquires additional stochastic Lie transport by the small scales. The stochastic dynamics of the composite homogenized flow is derived from a stochastic Hamilton’s principle and then recast into Lie–Poisson bracket form with a stochastic Hamiltonian. Several examples are given, including fluid flow with stochastically advected quantities and rigid body motion under gravity, i.e. the stochastic heavy top in a rotating frame.
Cotter CJ, Crisan D, Holm DD, et al., 2019, Numerically modelling stochastic lie transport in fluid dynamics, SIAM Journal on Scientific Computing, Vol: 17, Pages: 192-232, ISSN: 1064-8275
We present a numerical investigation of stochastic transport in ideal fluids.According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principlesof transformation theory and multi-time homogenisation, respectively, imply aphysically meaningful, data-driven approach for decomposing the fluid transportvelocity into its drift and stochastic parts, for a certain class of fluidflows. In the current paper, we develop new methodology to implement thisvelocity decomposition and then numerically integrate the resulting stochasticpartial differential equation using a finite element discretisation forincompressible 2D Euler fluid flows. The new methodology tested here is foundto be suitable for coarse graining in this case. Specifically, we performuncertainty quantification tests of the velocity decomposition of Cotter et al.(2017), by comparing ensembles of coarse-grid realisations of solutions of theresulting stochastic partial differential equation with the "true solutions" ofthe deterministic fluid partial differential equation, computed on a refinedgrid. The time discretization used for approximating the solution of thestochastic partial differential equation is shown to be consistent. We includecomprehensive numerical tests that confirm the non-Gaussianity of the streamfunction, velocity and vorticity fields in the case of incompressible 2D Eulerfluid flows.
Arnaudon A, Holm DD, Sommer S, 2019, Stochastic Metamorphosis with Template Uncertainties
In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex uids when the template properties are regarded as order parameters. The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.
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- Citations: 1
Holm DD, Tyranowski TM, 2018, Stochastic discrete Hamiltonian variational integrators, BIT Numerical Mathematics, Vol: 58, Pages: 1009-1048, ISSN: 0006-3835
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.
Crisan D, Holm DD, 2018, Wave breaking for the Stochastic Camassa-Holm equation, Physica D: Nonlinear Phenomena, Vol: 376-377, Pages: 138-143, ISSN: 0167-2789
We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.
Arnaudon A, Holm DD, Sommer S, 2018, A Geometric Framework for Stochastic Shape Analysis, Foundations of Computational Mathematics, ISSN: 1615-3375
We introduce a stochastic model of diffeomorphisms, whose action on a varietyof data types descends to stochastic evolution of shapes, images and landmarks.The stochasticity is introduced in the vector field which transports the datain the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework forshape analysis and image registration. The stochasticity thereby models errorsor uncertainties of the flow in following the prescribed deformation velocity.The approach is illustrated in the example of finite dimensional landmarkmanifolds, whose stochastic evolution is studied both via the Fokker-Planckequation and by numerical simulations. We derive two approaches for inferringparameters of the stochastic model from landmark configurations observed atdiscrete time points. The first of the two approaches matches moments of theFokker-Planck equation to sample moments of the data, while the second approachemploys an Expectation-Maximisation based algorithm using a Monte Carlo bridgesampling scheme to optimise the data likelihood. We derive and numerically testthe ability of the two approaches to infer the spatial correlation length ofthe underlying noise.
Arnaudon A, Holm D, Sommer S, 2018, String methods for stochastic image and shape matching, Journal of Mathematical Imaging and Vision, Vol: 60, Pages: 953-967, ISSN: 0924-9907
Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the large deformation diffeomorphic metric mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite-temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.
Gay-Balmaz F, Holm DD, 2018, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, Journal of Nonlinear Science, Vol: 28, Pages: 873-904, ISSN: 0938-8974
Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.
Holm DD, Tyranowski TM, 2018, New variational and multisymplectic formulations of the Euler-Poincare equation on the Virasoro-Bott group using the inverse map, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 474, ISSN: 1364-5021
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- Citations: 9
Holm DD, 2018, Stochastic metamorphosis in imaging science, ANNALS OF MATHEMATICAL SCIENCES AND APPLICATIONS, Vol: 3, Pages: 309-335, ISSN: 2380-288X
Arnaudon A, Ganaba N, Holm D, 2018, The Stochastic Energy-Casimir Method, Comptes Rendus Mécanique, ISSN: 1631-0721
In this paper, we extend the energy-Casimir stability method fordeterministic Lie-Poisson Hamiltonian systems to provide sufficient conditionsfor the stability in probability of stochastic dynamical systems withsymmetries and multiplicative noise. We illustrate this theory with classicalexamples of coadjoint motion, including the rigid body, the heavy top and thecompressible Euler equation in two dimensions. The main result of thisextension is that stable deterministic equilibria remain stable in probabilityup to a certain stopping time which depends on the amplitude of the noise forfinite dimensional systems and on the amplitude the spatial derivative of thenoise for infinite dimensional systems.
Holm DD, Putkaradze V, 2018, Dynamics of non-holonomic systems with stochastic transport, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 474, ISSN: 1364-5021
Cruzeiro AB, Holm DD, Ratiu TS, 2018, Momentum Maps and Stochastic Clebsch Action Principles, COMMUNICATIONS IN MATHEMATICAL PHYSICS, Vol: 357, Pages: 873-912, ISSN: 0010-3616
Ribeiro Castro A, 2017, Noise and dissipation in rigid body motion, Publisher: Springer
Using the rigid body as an example, we illustrate some features of stochastic geometric mechanics. These features include: (i) a geometric variational motivation for the noise structure involving Lie-Poisson brackets and momentum maps , (ii) stochastic coadjoint motion with double bracket dissipation , (iii) description and its stationary solutions , (iv) random dynamical systems , random attractors and SRB measures connected to statistical physics.
Cotter CJ, Gottwald G, Holm DD, 2017, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 473, ISSN: 1364-5021
In Holm (Holm 2015 Proc. R. Soc. A 471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
Arnaudon A, Lopez MC, Holm D, 2017, Un-reduction in field theory, Letters in Mathematical Physics, Vol: 108, Pages: 225-247, ISSN: 0377-9017
The un-reduction procedure introduced previously in the context of Mechanicsis extended to covariant Field Theory. The new covariant un-reduction procedureis applied to the problem of shape matching of images which depend on more thanone independent variable (for instance, time and an additional labellingparameter). Other possibilities are also explored: non-linear $\sigma$-modelsand the hyperbolic flows of curves.
Arnaudon A, De Castro AL, Holm D, 2017, Noise and Dissipation on Coadjoint Orbits, Journal of Nonlinear Science, Vol: 28, Pages: 91-145, ISSN: 0938-8974
We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.
Holm DD, Jacobs HO, 2017, Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics, Journal of Nonlinear Science, Vol: 27, Pages: 973-1006, ISSN: 0938-8974
Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.
Arnaudon A, Holm DD, Pai A, et al., 2017, A stochastic large deformation model for computational anatomy, Information Processing in Medical Imaging
In the study of shapes of human organs using computational anatomy, variations are found to arise from inter-subject anatomical differences, disease-specific effects, and measurement noise. This paper introduces a stochastic model for incorporating random variations into the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. By accounting for randomness in a particular setup which is crafted to fit the geometrical properties of LDDMM, we formulate the template estimation problem for landmarks with noise and give two methods for efficiently estimating the parameters of the noise fields from a prescribed data set. One method directly approximates the time evolution of the variance of each landmark by a finite set of differential equations, and the other is based on an Expectation-Maximisation algorithm. In the second method, the evaluation of the data likelihood is achieved without registering the landmarks, by applying bridge sampling using a stochastically perturbed version of the large deformation gradient flow algorithm. The method and the estimation algorithms are experimentally validated on synthetic examples and shape data of human corpora callosa.
Gibbon JD, Holm DD, 2017, Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations, Nonlinearity, Vol: 30, Pages: R1-R24, ISSN: 0951-7715
We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members ${{ \Omega }_{m}}(t)$ ($1\leqslant m<\infty $ ) are made up from the respective sum of the L 2m -norms of vorticity and the density gradient. Each ${{ \Omega }_{m}}(t)$ has a lower bound in terms of the inverse Rossby number, Ro −1, that turns out to be crucial to the argument. For convenience, the ${{ \Omega }_{m}}$ are also scaled into a new set of variables D m (t). By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the D m (t) in terms of Ro −1 and the Reynolds number Re. These upper bounds vary across bands in the $\left\{{{D}_{1}},\,{{D}_{m}}\right\}$ phase plane. The boundaries of these bands depend subtly upon Ro −1, Re, and the inverse Froude number Fr −1. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${{ \Omega }_{1}}$ deviates from Re 3/4 as a function of $R{{o}^{-1}},\,Re$ and Fr −1.
Arnaudon A, Holm DD, Ivanov RI, 2017, G-Strands on symmetric spaces., Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, Vol: 473, ISSN: 1471-2946
We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S(1) and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa-Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.
Albeverio S, Cruzeiro AB, Holm D, 2017, Preface, Pages: v-x, ISSN: 2194-1009
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Cotter CJ, Eldering J, Holm DD, et al., 2016, Weak dual pairs and jetlet methods for ideal incompressible fluid models in n >= 2 dimensions, Journal of Nonlinear Science, Vol: 26, Pages: 1723-1765, ISSN: 1432-1467
We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin’s circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n≥2n≥2 and which exhibit a shadow of Kelvin’s circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.
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