Darryl D. Holm

Darryl D. Holm, Tomasz M. Tyranowski

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.

Alexis Arnaudon, Darryl D. Holm, Stefan Sommer

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker-Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker-Planck equation to sample moments of the data, while the second approach employs an Expectation-Maximisation based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

Alexis Arnaudon, Darryl D. Holm, Akshay Pai et al.

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.

Alexis Arnaudon, Marco Castrillon Lopez, Darryl D. Holm

The process of un-reduction, a sort of reversal of reduction by the Lie group symmetries of a variational problem, is explored in the setting of field theories. This process is applied to the problem of curve matching in the plane, when the curves depend on more than one independent variable. This situation occurs in a variety of instances such as matching of surfaces or comparison of evolution between species. A discussion of the appropriate Lagrangian involved in the variational principle is given, as well as some initial numerical investigations.

Bernard J. Geurts, Darryl D. Holm

We consider so-called Leray regularization of the convective contributions. This gives rise to a subgrid parameterization which involves both explicit filtering and (approximate) inversion. The Leray model also arises from the alpha-modeling strategy derived via Kelvin's circulation theorem. We study the dynamics associated with the Leray model in a turbulent mixing layer and compare predictions with filtered DNS results and findings due to dynamic (mixed) models. In particular, the kinetic energy, momentum thickness and energy-spectra are analyzed, establishing favorable performance of the Leray model and robustness at arbitrarily high Reynolds number. This is unique for a similarity-type model that does not contain an explicit eddy-viscosity term.