Publications
42 results found
Arnaudon A, López MC, Holm DD, 2018, Un-reduction in field theory, Letters in Mathematical Physics, Vol: 108, Pages: 225-247, ISSN: 0377-9017
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.
Arnaudon A, Gibbon JD, 2017, Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates, Physical Review A, Vol: 96, ISSN: 2469-9926
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.
Sommer S, Arnaudon A, Kuhnel L, et al., 2017, Bridge simulation and metric estimation on landmark manifolds, Information Processing in Medical Imaging 2017, Publisher: Springer Verlag, Pages: 571-582, ISSN: 0302-9743
We present an inference algorithm and connected Monte Carlo based estimationprocedures for metric estimation from landmark configurations distributedaccording to the transition distribution of a Riemannian Brownian motionarising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric.The distribution possesses properties similar to the regular Euclidean normaldistribution but its transition density is governed by a high-dimensional PDEwith no closed-form solution in the nonlinear case. We show how the density canbe numerically approximated by Monte Carlo sampling of conditioned Brownianbridges, and we use this to estimate parameters of the LDDMM kernel and thusthe metric structure by maximum likelihood.
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.
Sommer S, Arnaudon A, Kuhnel L, et al., 2017, Bridge Simulation and Metric Estimation on Landmark Manifolds, 1st International Workshop on Graphs in Biomedical Image Analysis (GRAIL) / 6th International Workshop on Mathematical Foundations of Computational Anatomy (MFCA) / 3rd International Workshop on Imaging Genetics (MICGen), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 79-91, ISSN: 0302-9743
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- Citations: 11
Arnaudon A, 2016, On a lagrangian reduction and a deformation of completely integrable systems, Journal of Nonlinear Science, Vol: 26, Pages: 1133-1160, ISSN: 1432-1467
We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm H1H1 in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered, and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them, we found two important equations, the Camassa–Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation.
Revaz Y, Arnaudon A, Nichols M, et al., 2016, Computational issues in chemo-dynamical modelling of the formation and evolution of galaxies, Astronomy & Astrophysics, Vol: 588, ISSN: 1432-0746
Chemo-dynamical N-body simulations are an essential tool for understanding the formation and evolution of galaxies. As the number of observationally determined stellar abundances continues to climb, these simulations are able to provide new constraints on the early star formaton history and chemical evolution inside both the Milky Way and Local Group dwarf galaxies. Here, we aim to reproduce the low α-element scatter observed in metal-poor stars. We first demonstrate that as stellar particles inside simulations drop below a mass threshold, increases in the resolution produce an unacceptably large scatter as one particle is no longer a good approximation of an entire stellar population. This threshold occurs at around 103M⊙, a mass limit easily reached in current (and future) simulations. By simulating the Sextans and Fornax dwarf spheroidal galaxies we show that this increase in scatter at high resolutions arises from stochastic supernovae explosions. In order to reduce this scatter down to the observed value, we show the necessity of introducing a metal mixing scheme into particle-based simulations. The impact of the method used to inject the metals into the surrounding gas is also discussed. We finally summarise the best approach for accurately reproducing the scatter in simulations of both Local Group dwarf galaxies and in the Milky Way.
Arnaudon A, 2016, On a deformation of the nonlinear Schrödinger equation, Journal of Physics A: Mathematical and Theoretical, Vol: 49, ISSN: 1751-8113
We study a deformation of the nonlinear Schrödinger (NLS) equation recently derived in the context of deformation of hierarchies of integrable systems. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions which can be smooth or peaked, then with the help of numerical simulations we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly, the structure of the solution during the collision of solitary waves or during the rogue wave events is sharper and has larger amplitudes than in the classical NLS equation.
Arnaudon A, 2015, The stochastic integrable AKNS hierarchy
We derive a stochastic AKNS hierarchy using geometrical methods. Theintegrability is shown via a stochastic zero curvature relation associated witha stochastic isospectral problem. We expose some of the stochastic integrablepartial differential equations which extend the stochastic KdV equationdiscovered by M. Wadati in 1983 for all the AKNS flows. We also show how tofind stochastic solitons from the stochastic evolution of the scattering dataof the stochastic IST. We finally expose some properties of these equations andalso briefly study a stochastic Camassa-Holm equation which reduces to astochastic Hamiltonian system of peakons.
Arnaudon A, Lopez MC, Holm DD, 2015, Covariant un-reduction for curve matching, MFCA 2015, Publisher: MICCAI Society, Pages: 95-106
The process of un-reduction, a sort of reversal of reduction by the Lie groupsymmetries of a variational problem, is explored in the setting of fieldtheories. This process is applied to the problem of curve matching in theplane, when the curves depend on more than one independent variable. Thissituation occurs in a variety of instances such as matching of surfaces orcomparison of evolution between species. A discussion of the appropriateLagrangian involved in the variational principle is given, as well as someinitial numerical investigations.
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