62,977 research outputs found
Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems
Model order reduction provides low-complexity high-fidelity surrogate models
that allow rapid and accurate solutions of parametric differential equations.
The development of reduced order models for parametric nonlinear Hamiltonian
systems is still challenged by several factors: (i) the geometric structure
encoding the physical properties of the dynamics; (ii) the slowly decaying
Kolmogorov -width of conservative dynamics; (iii) the gradient structure of
the nonlinear flow velocity; (iv) high variations in the numerical rank of the
state as a function of time and parameters. We propose to address these aspects
via a structure-preserving adaptive approach that combines symplectic dynamical
low-rank approximation with adaptive gradient-preserving hyper-reduction and
parameters sampling. Additionally, we propose to vary in time the dimensions of
both the reduced basis space and the hyper-reduction space by monitoring the
quality of the reduced solution via an error indicator related to the
projection error of the Hamiltonian vector field. The resulting adaptive
hyper-reduced models preserve the geometric structure of the Hamiltonian flow,
do not rely on prior information on the dynamics, and can be solved at a cost
that is linear in the dimension of the full order model and linear in the
number of test parameters. Numerical experiments demonstrate the improved
performances of the resulting fully adaptive models compared to the original
and reduced order models
Reducing "Structure From Motion": a General Framework for Dynamic Vision - Part 1: Modeling
The literature on recursive estimation of structure and motion from monocular image sequences comprises a large number of different models and estimation techniques. We propose a framework that allows us to derive and compare all models by following the idea of dynamical system reduction.
The "natural" dynamic model, derived by the rigidity constraint and the perspective projection, is first reduced by explicitly decoupling structure (depth) from motion. Then implicit decoupling techniques are explored, which consist of imposing that some function of the unknown parameters is held constant. By appropriately choosing such a function, not only can we account for all models seen so far in the literature, but we can also derive novel ones
On singular Lagrangians affine in velocities
The properties of Lagrangians affine in velocities are analyzed in a
geometric way. These systems are necessarily singular and exhibit, in general,
gauge invariance. The analysis of constraint functions and gauge symmetry leads
us to a complete classification of such Lagrangians.Comment: AMSTeX, 22 page
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