114 research outputs found
Balanced data assimilation for highly-oscillatory mechanical systems
Data assimilation algorithms are used to estimate the states of a dynamical
system using partial and noisy observations. The ensemble Kalman filter has
become a popular data assimilation scheme due to its simplicity and robustness
for a wide range of application areas. Nevertheless, the ensemble Kalman filter
also has limitations due to its inherent Gaussian and linearity assumptions.
These limitations can manifest themselves in dynamically inconsistent state
estimates. We investigate this issue in this paper for highly oscillatory
Hamiltonian systems with a dynamical behavior which satisfies certain balance
relations. We first demonstrate that the standard ensemble Kalman filter can
lead to estimates which do not satisfy those balance relations, ultimately
leading to filter divergence. We also propose two remedies for this phenomenon
in terms of blended time-stepping schemes and ensemble-based penalty methods.
The effect of these modifications to the standard ensemble Kalman filter are
discussed and demonstrated numerically for two model scenarios. First, we
consider balanced motion for highly oscillatory Hamiltonian systems and,
second, we investigate thermally embedded highly oscillatory Hamiltonian
systems. The first scenario is relevant for applications from meteorology while
the second scenario is relevant for applications of data assimilation to
molecular dynamics
Optimization on manifolds: A symplectic approach
There has been great interest in using tools from dynamical systems and
numerical analysis of differential equations to understand and construct new
optimization methods. In particular, recently a new paradigm has emerged that
applies ideas from mechanics and geometric integration to obtain accelerated
optimization methods on Euclidean spaces. This has important consequences given
that accelerated methods are the workhorses behind many machine learning
applications. In this paper we build upon these advances and propose a
framework for dissipative and constrained Hamiltonian systems that is suitable
for solving optimization problems on arbitrary smooth manifolds. Importantly,
this allows us to leverage the well-established theory of symplectic
integration to derive "rate-matching" dissipative integrators. This brings a
new perspective to optimization on manifolds whereby convergence guarantees
follow by construction from classical arguments in symplectic geometry and
backward error analysis. Moreover, we construct two dissipative generalizations
of leapfrog that are straightforward to implement: one for Lie groups and
homogeneous spaces, that relies on the tractable geodesic flow or a retraction
thereof, and the other for constrained submanifolds that is based on a
dissipative generalization of the famous RATTLE integrator
Multi-symplectic discretisation of wave map equations
We present a new multi-symplectic formulation of constrained Hamiltonian
partial differential equations, and we study the associated local conservation
laws. A multi-symplectic discretisation based on this new formulation is
exemplified by means of the Euler box scheme. When applied to the wave map
equation, this numerical scheme is explicit, preserves the constraint and can
be seen as a generalisation of the Shake algorithm for constrained mechanical
systems. Furthermore, numerical experiments show excellent conservation
properties of the numerical solutions
Discrete mechanics and variational integrators
This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented
Optimization via conformal Hamiltonian systems on manifolds
In this work we propose a method to perform optimization on manifolds. We
assume to have an objective function defined on a manifold and think of it
as the potential energy of a mechanical system. By adding a momentum-dependent
kinetic energy we define its Hamiltonian function, which allows us to write the
corresponding Hamiltonian system. We make it conformal by introducing a
dissipation term: the result is the continuous model of our scheme. We solve it
via splitting methods (Lie-Trotter and leapfrog): we combine the RATTLE scheme,
approximating the conserved flow, with the exact dissipated flow. The result is
a conformal symplectic method for constant stepsizes. We also propose an
adaptive stepsize version of it. We test it on an example, the minimization of
a function defined on a sphere, and compare it with the usual gradient descent
method.Comment: 21 pages, 6 figures, 1 page. Presented at GSI conference 202
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
Stochastic Variational Integrators
This paper presents a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. The main result is to derive
stochastic governing equations for such systems from a critical point of a
stochastic action. Using this result the paper derives Langevin-type equations
for constrained mechanical systems and implements a stochastic analog of
Lagrangian reduction. These are easy consequences of the fact that the
stochastic action is intrinsically defined. Stochastic variational integrators
(SVIs) are developed using a discretized stochastic variational principle. The
paper shows that the discrete flow of an SVI is a.s. symplectic and in the
presence of symmetry a.s. momentum-map preserving. A first-order mean-square
convergent SVI for mechanical systems on Lie groups is introduced. As an
application of the theory, SVIs are exhibited for multiple, randomly forced and
torqued rigid-bodies interacting via a potential.Comment: 21 pages, 8 figure
Nonholonomic Dynamics
Nonholonomic systems are, roughly speaking, mechanical
systems with constraints on their velocity
that are not derivable from position constraints.
They arise, for instance, in mechanical systems
that have rolling contact (for example, the rolling
of wheels without slipping) or certain kinds of sliding
contact (such as the sliding of skates). They are
a remarkable generalization of classical Lagrangian
and Hamiltonian systems in which one allows position
constraints only.
There are some fascinating differences between
nonholonomic systems and classical Hamiltonian
or Lagrangian systems. Among other things: nonholonomic
systems are nonvariational—they arise
from the Lagrange-d’Alembert principle and not
from Hamilton’s principle; while energy is preserved
for nonholonomic systems, momentum is
not always preserved for systems with symmetry
(i.e., there is nontrivial dynamics associated with
the nonholonomic generalization of Noether’s
theorem); nonholonomic systems are almost Poisson
but not Poisson (i.e., there is a bracket that together
with the energy on the phase space defines
the motion, but the bracket generally does not satisfy
the Jacobi identity); and finally, unlike the
Hamiltonian setting, volume may not be preserved
in the phase space, leading to interesting asymptotic
stability in some cases, despite energy conservation.
The purpose of this article is to engage
the reader’s interest by highlighting some of these
differences along with some current research in the
area. There has been some confusion in the literature
for quite some time over issues such as the
variational character of nonholonomic systems, so
it is appropriate that we begin with a brief review
of the history of the subject
Asynchronous Variational Contact Mechanics
An asynchronous, variational method for simulating elastica in complex
contact and impact scenarios is developed. Asynchronous Variational Integrators
(AVIs) are extended to handle contact forces by associating different time
steps to forces instead of to spatial elements. By discretizing a barrier
potential by an infinite sum of nested quadratic potentials, these extended
AVIs are used to resolve contact while obeying momentum- and
energy-conservation laws. A series of two- and three-dimensional examples
illustrate the robustness and good energy behavior of the method
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