## Publications

335 results found

Crisan D, Holm DD, Street OD, 2021, Wave-current interaction on a free surface, *STUDIES IN APPLIED MATHEMATICS*, ISSN: 0022-2526

Holm DD, Hu R, 2021, Stochastic effects of waves on currents in the ocean mixed layer, *Journal of Mathematical Physics*, Vol: 62, ISSN: 0022-2488

This paper introduces an energy-preserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics and the Euler-Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. We also present an Eulerian-averaged SFLT model (EA SFLT), based on decomposing the Eulerian solutions of the energy-conserving SFLT model into sums of their expectations and fluctuations.

Bendall TM, Cotter CJ, Holm DD, 2021, Perspectives on the formation of peakons in the stochastic Camassa-Holm equation, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 477, ISSN: 1364-5021

Holm D, Luesink E, Pan W, 2021, Stochastic mesoscale circulation dynamics in the thermal ocean, *Physics of Fluids*, Vol: 33, Pages: 1-22, ISSN: 1070-6631

In analogy with similar effects in adiabatic compressible fluid dynamics, the effects of buoyancy gradients on incompressible stratified flows are said to be “thermal.” The thermal rotating shallow water (TRSW) model equations contain three small nondimensional parameters. These are the Rossby number, the Froude number, and the buoyancy parameter. Asymptotic expansion of the TRSW model equations in these three small parameters leads to the deterministic thermal versions of the Salmon’s L1 (TL1) model and the thermal quasi-geostrophic (TQG) model, upon expanding in the neighborhood of thermal quasi-geostrophic balance among the flow velocity and the gradients of free surface elevation and buoyancy. The linear instability of TQG at high wavenumber tends to create circulation at small scales. Such a high- wavenumber instability could be unresolvable in many computational simulations, but its presence at small scales may contribute signifi- cantly to fluid transport at resolvable scales. Sometimes, such effects are modeled via “stochastic backscatter of kinetic energy.” Here, we try another approach. Namely, we model “stochastic transport” in the hierarchy of models TRSW/TL1/TQG. The models are derived via the approach of stochastic advection by Lie transport (SALT) as obtained from a recently introduced stochastic version of the Euler–Poincare var- iational principle. We also indicate the potential next steps for applying these models in uncertainty quantification and data assimilation of the rapid, high-wavenumber effects of buoyancy fronts at these three levels of description by using the data-driven stochastic parametrization algorithms derived previously using the SALT approach.

Holm DD, Luesink E, 2021, Stochastic Wave-Current Interaction in Thermal Shallow Water Dynamics, *JOURNAL OF NONLINEAR SCIENCE*, Vol: 31, ISSN: 0938-8974

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- Citations: 1

Holm DD, 2021, Stochastic Variational Formulations of Fluid Wave-Current Interaction, *JOURNAL OF NONLINEAR SCIENCE*, Vol: 31, ISSN: 0938-8974

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- Citations: 1

Drivas TD, Holm D, 2020, Circulation and energy theorem preserving stochastic fluids, *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, Vol: 150, Pages: 2776-2814, ISSN: 0308-2105

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem [1]. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin’s theorem, introduced by Constantin–Iyer (2008) [3]. In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincare´ and stochastic Navier-Stokes–Poincare´ equations respectively. The stochastic Euler–Poincare equations were previously derived from a stochastic ´ variational principle by Holm (2015) [20], which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Cotter C, Crisan D, Holm DD, et al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Geurts BJ, Holm DD, Luesink E, 2020, Lyapunov Exponents of Two Stochastic Lorenz 63 Systems, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1343-1365, ISSN: 0022-4715

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- Citations: 3

Alonso-Oran D, de Leon AB, Holm DD,
et al., 2020, Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler-Boussinesq Equation with Transport Noise, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1267-1303, ISSN: 0022-4715

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- Citations: 3

Holm DD, 2020, Stochastic modelling in fluid dynamics: Ito versus Stratonovich, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 476, ISSN: 1364-5021

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- Citations: 1

Holm D, 2020, Stochastic modelling in fluid dynamics: It\^o vs stratonovich, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, Pages: 1-12, ISSN: 1364-5021

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.

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

Cotter C, Crisan D, Holm D, et al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

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- Citations: 7

de Leon AB, Holm DD, Luesink E, et al., 2020, Implications of Kunita-Ito-Wentzell Formula for k-Forms in Stochastic Fluid Dynamics, Publisher: SPRINGER

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- Citations: 5

Drivas TD, Holm DD, Leahy J-M, 2020, Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1304-1342, ISSN: 0022-4715

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- Citations: 4

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

Holm DD, 2019, Stochastic Closures for Wave-Current Interaction Dynamics, Publisher: SPRINGER

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- Citations: 7

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.

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.

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