233 research outputs found
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
Krylov projection methods for linear Hamiltonian systems
We study geometric properties of Krylov projection methods for large and
sparse linear Hamiltonian systems. We consider in particular energy
preservation. We discuss the connection to structure preserving model
reduction. We illustrate the performance of the methods by applying them to
Hamiltonian PDEs.Comment: 16 pages, 17 figure
Passivity-preserving splitting methods for rigid body systems
A rigid body model for the dynamics of a marine vessel, used in simulations
of offshore pipe-lay operations, gives rise to a set of ordinary differential
equations with controls. The system is input-output passive. We propose
passivity-preserving splitting methods for the numerical solution of a class of
problems which includes this system as a special case. We prove the
passivity-preservation property for the splitting methods, and we investigate
stability and energy behaviour in numerical experiments. Implementation is
discussed in detail for a special case where the splitting gives rise to the
subsequent integration of two completely integrable flows. The equations for
the attitude are reformulated on using rotation matrices rather than
local parametrizations with Euler angles.Comment: 27 pages, 4 figures. To be published in 'Multibody System Dynamics
Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
This paper concerns an extension of discrete gradient methods to
finite-dimensional Riemannian manifolds termed discrete Riemannian gradients,
and their application to dissipative ordinary differential equations. This
includes Riemannian gradient flow systems which occur naturally in optimization
problems. The Itoh--Abe discrete gradient is formulated and applied to gradient
systems, yielding a derivative-free optimization algorithm. The algorithm is
tested on two eigenvalue problems and two problems from manifold valued
imaging: InSAR denoising and DTI denoising.Comment: Post-revision version. To appear in SIAM Journal on Scientific
Computin
Shape analysis on Lie groups and homogeneous spaces
In this paper we are concerned with the approach to shape analysis based on
the so called Square Root Velocity Transform (SRVT). We propose a
generalisation of the SRVT from Euclidean spaces to shape spaces of curves on
Lie groups and on homogeneous manifolds. The main idea behind our approach is
to exploit the geometry of the natural Lie group actions on these spaces.Comment: 8 pages, Contribution to the conference "Geometric Science of
Information '17
A novel approach to rigid spheroid models in viscous flows using operator splitting methods
Calculating cost-effective solutions to particle dynamics in viscous flows is
an important problem in many areas of industry and nature. We implement a
second-order symmetric splitting method on the governing equations for a rigid
spheroidal particle model with torques, drag and gravity. The method splits the
operators into a vector field that is conservative and one that takes into
account the forces of the fluid. Error analysis and numerical tests are
performed on perturbed and stiff particle-fluid systems. For the perturbed
case, the splitting method greatly improves the solution accuracy, when
compared to a conventional multi-step method, and the global error behaves as
for roughly equal computational cost. For stiff
systems, we show that the splitting method retains stability in regimes where
conventional methods blow up. In addition, we show through numerical
experiments that the global order is reduced from
in the non-stiff regime to in
the stiff regime.Comment: 24 pages, 6 figures (13 if you count sub figs), all figures are in
colou
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