Publications
92 results found
Melvin T, Benacchio T, Shipway B, et al., 2019, A mixed finite-element, finite-volume, semi-implicit discretisation for atmospheric dynamics: Cartesian geometry, Quarterly Journal of the Royal Meteorological Society, ISSN: 0035-9009
Cotter C, Cotter S, Russell P, Ensemble transport adaptive importance sampling, SIAM/ASA Journal on Uncertainty Quantification, ISSN: 2166-2525
Markov chain Monte Carlo methods are a powerful and commonly used family ofnumerical methods for sampling from complex probability distributions. As applications of thesemethods increase in size and complexity, the need for efficient methods increases. In this paper, wepresent a particle ensemble algorithm. At each iteration, an importance sampling proposal distri-bution is formed using an ensemble of particles. A stratified sample is taken from this distributionand weighted under the posterior, a state-of-the-art ensemble transport resampling method is thenused to create an evenly weighted sample ready for the next iteration. We demonstrate that thisensemble transport adaptive importance sampling (ETAIS) method outperforms MCMC methodswith equivalent proposal distributions for low dimensional problems, and in fact shows better thanlinear improvements in convergence rates with respect to the number of ensemble members. We alsointroduce a new resampling strategy, multinomial transformation (MT), which while not as accurateas the ensemble transport resampler, is substantially less costly for large ensemble sizes, and canthen be used in conjunction with ETAIS for complex problems. We also focus on how algorithmicparameters regarding the mixture proposal can be quickly tuned to optimise performance. In partic-ular, we demonstrate this methodology’s superior sampling for multimodal problems, such as thosearising from inference for mixture models, and for problems with expensive likelihoods requiring thesolution of a differential equation, for which speed-ups of orders of magnitude are demonstrated.Likelihood evaluations of the ensemble could be computed in a distributed manner, suggesting thatthis methodology is a good candidate for parallel Bayesian computations.
Cotter CJ, Crisan D, Holm DD, et al., Numerically Modelling Stochastic Lie Transport in Fluid Dynamics, SIAM Journal on Scientific Computing, 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.
Budd CJ, McRae ATT, Cotter CJ, 2018, The scaling and skewness of optimally transported meshes on the sphere, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 375, Pages: 540-564, ISSN: 0021-9991
Shipton J, Gibson TH, Cotter CJ, 2018, Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere, Journal of Computational Physics, Vol: 375, Pages: 1121-1137, ISSN: 0021-9991
We describe a compatible finite element discretisation for the shallow waterequations on the rotating sphere, concentrating on integrating consistentupwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potentialvorticity that satisfies a stable upwinded advection equation through aTaylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.
Ham D, Mitchell L, Gibson T, et al., 2018, firedrakeproject/firedrake: an automated finite element system
This release is specifically created to document the version of firedrake used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html
Ham D, Homolya M, Kirby R, et al., 2018, firedrakeproject/fiat: The Finite Element Automated Tabulator
This release is specifically created to document the version of fiat used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html
Bendall TM, Cotter CJ, Statistical properties of an enstrophy conserving discretisation for the stochastic quasi-geostrophic equation, Geophysical and Astrophysical Fluid Dynamics, ISSN: 0309-1929
A framework of variational principles for stochastic fluid dynamics was presented by Holm (2015), and these stochastic equations were also derived by Cotter et al. (2017). We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank (2007), who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is agreement between the statistics under a wide range of set-ups.
Bauer W, Cotter CJ, 2018, Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions, Journal of Computational Physics, Vol: 373, Pages: 171-187, ISSN: 0021-9991
We describe an energy-enstrophy conserving discretisation for the rotatingshallow water equations with slip boundary conditions. This relaxes theassumption of boundary-free domains (periodic solutions or the surface of asphere, for example) in the energy-enstrophy conserving formulation of McRaeand Cotter (2014). This discretisation requires extra prognostic vorticityvariables on the boundary in addition to the prognostic velocity and layerdepth variables. The energy-enstrophy conservation properties hold for anyappropriate set of compatible finite element spaces defined on arbitrary mesheswith arbitrary boundaries. We demonstrate the conservation properties of thescheme with numerical solutions on a rotating hemisphere.
Cotter CJ, Cullen MJP, Particle relabelling symmetries and Noether's theorem for vertical slice models, Journal of Geometric Mechanics, ISSN: 1941-4889
We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc RoySoc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closedloops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can bederived directly from this circulation theorem. In this paper, we show that this property is due to these modelshaving a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, notjust those diffeomorphisms that preserve the potential temperature. This is developed using the methodologyof Cotter and Holm (Foundations of Computational Mathematics, 2012).
Cotter CJ, Graber PJ, Kirby RC, 2018, Mixed finite elements for global tide models with nonlinear damping, Numerische Mathematik, ISSN: 0029-599X
We study mixed finite element methods for the rotating shallow waterequations with linearized momentum terms but nonlinear drag. By means of anequivalent second-order formulation, we prove long-time stability of the systemwithout energy accumulation. We also give rates of damping in unforced systemsand various continuous dependence results on initial conditions and forcingterms. \emph{A priori} error estimates for the momentum and free surfaceelevation are given in $L^2$ as well as for the time derivative and divergenceof the momentum. Numerical results confirm the theoretical results regardingboth energy damping and convergence rates.
Natale A, Cotter CJ, 2018, A variational H(div) finite-element discretization approach for perfect incompressible fluids, IMA JOURNAL OF NUMERICAL ANALYSIS, Vol: 38, Pages: 1388-1419, ISSN: 0272-4979
- Author Web Link
- Open Access Link
- Cite
- Citations: 2
Melvin T, Benacchio T, Thuburn J, et al., 2018, Choice of function spaces for thermodynamic variables in mixed finite-element methods, QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Vol: 144, Pages: 900-916, ISSN: 0035-9009
Natale A, Cotter CJ, 2018, Corrigendum to: A variational H(div) finite-element discretization approach for perfect incompressible fluids, IMA Journal of Numerical Analysis, Vol: 38, Pages: 1084-1084, ISSN: 0272-4979
This is a correction to:IMA Journal of Numerical Analysis, Volume 38, Issue 3, 17 July 2018, Pages 1388–1419, https://doi.org/10.1093/imanum/drx033
McRae ATT, Cotter CJ, Budd CJ, 2018, Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements, SIAM Journal on Scientific Computing, Vol: 40, Pages: A1121-A1148
In moving mesh methods, the underlying mesh is dynamically adapted withoutchanging the connectivity of the mesh. We specifically consider the generationof meshes which are adapted to a scalar monitor function throughequidistribution. Together with an optimal transport condition, this leads to aMonge-Amp\`ere equation for a scalar mesh potential. We adapt an existingfinite element scheme for the standard Monge-Amp\`ere equation to this meshgeneration problem; this is a mixed finite element scheme, in which an extradiscrete variable is introduced to represent the Hessian matrix of secondderivatives. The problem we consider has additional nonlinearities over thebasic Monge-Amp\`ere equation due to the implicit dependence of the monitorfunction on the resulting mesh. We also derive the equivalentMonge-Amp\`ere-like equation for generating meshes on the sphere. The finiteelement scheme is extended to the sphere, and we provide numerical examples.All numerical experiments are performed using the open-source finite elementframework Firedrake.
van Sebille E, Griffies SM, Abernathey R, et al., 2018, Lagrangian ocean analysis: Fundamentals and practices, OCEAN MODELLING, Vol: 121, Pages: 49-75, ISSN: 1463-5003
- Author Web Link
- Open Access Link
- Cite
- Citations: 21
Cotter CJ, Gottwald GA, 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
- Author Web Link
- Open Access Link
- Cite
- Citations: 9
Yamazaki H, Shipton J, Cullen MJP, et al., 2017, Vertical slice modelling of nonlinear Eady waves using a compatible finite element method, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 343, Pages: 130-149, ISSN: 0021-9991
- Author Web Link
- Open Access Link
- Cite
- Citations: 1
Natale A, Cotter CJ, 2017, Scale-selective dissipation in energy-conserving finite-element schemes for two-dimensional turbulence, QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Vol: 143, Pages: 1734-1745, ISSN: 0035-9009
- Author Web Link
- Open Access Link
- Cite
- Citations: 2
Gregory A, Cotter CJ, 2017, On the calibration of multilevel Monte Carlo ensemble forecasts, QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Vol: 143, Pages: 1929-1935, ISSN: 0035-9009
Gregory A, Cotter CJ, 2017, A SEAMLESS MULTILEVEL ENSEMBLE TRANSFORM PARTICLE FILTER, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 39, Pages: A2684-A2701, ISSN: 1064-8275
- Author Web Link
- Open Access Link
- Cite
- Citations: 1
Abolghasemi MA, Piggott MD, Spinneken J, et al., 2016, Simulating tidal turbines with multi-scale mesh optimisation techniques, JOURNAL OF FLUIDS AND STRUCTURES, Vol: 66, Pages: 69-90, ISSN: 0889-9746
- Author Web Link
- Open Access Link
- Cite
- Citations: 8
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.
Cotter CJ, Kirby RC, 2016, Mixed finite elements for global tide models, NUMERISCHE MATHEMATIK, Vol: 133, Pages: 255-277, ISSN: 0029-599X
- Author Web Link
- Open Access Link
- Cite
- Citations: 1
Cotter CJ, Kuzmin D, 2016, Embedded discontinuous Galerkin transport schemes with localised limiters, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 311, Pages: 363-373, ISSN: 0021-9991
- Author Web Link
- Open Access Link
- Cite
- Citations: 2
Gregory A, Cotter CJ, Reich S, 2016, MULTILEVEL ENSEMBLE TRANSFORM PARTICLE FILTERING, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 38, Pages: A1317-A1338, ISSN: 1064-8275
- Author Web Link
- Open Access Link
- Cite
- Citations: 7
Natale A, Shipton J, Cotter CJ, 2016, Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Vol: 1
Mcrae ATT, Bercea G-T, Mitchell L, et al., 2016, AUTOMATED GENERATION AND SYMBOLIC MANIPULATION OF TENSOR PRODUCT FINITE ELEMENTS, SIAM JOURNAL ON SCIENTIFIC COMPUTING, Vol: 38, Pages: S25-S47, ISSN: 1064-8275
- Author Web Link
- Open Access Link
- Cite
- Citations: 9
Jordi BE, Cotter CJ, Sherwin SJ, 2015, An adaptive selective frequency damping method, PHYSICS OF FLUIDS, Vol: 27, ISSN: 1070-6631
- Author Web Link
- Open Access Link
- Cite
- Citations: 5
Reich S, Cotter CJ, 2015, Probabilistic Forecasting and Bayesian Data Assimilation, http://www.cambridge.org/us/academic/subjects/mathematics/computational-science/probabilistic-forecasting-and-bayesian-data-assimilation, Publisher: Cambridge University Press, ISBN: 9781107663916
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.