909 research outputs found
Asymptotic and Lyapunov stability of Poisson equilibria
This paper includes results centered around three topics, all of them related
with the nonlinear stability of equilibria in Poisson dynamical systems.
Firstly, we prove an energy-Casimir type sufficient condition for stability
that uses functions that are not necessarily conserved by the flow and that
takes into account certain asymptotically stable behavior that may occur in the
Poisson category. This method is adapted to Poisson systems obtained via a
reduction procedure and we show in examples that the kind of stability that we
propose is appropriate when dealing with the stability of the equilibria of
some constrained systems. Finally, we discuss two situations in which the use
of continuous Casimir functions in stability studies is equivalent to the
topological stability methods introduced by Patrick {\it et al.}Comment: 23 pages, 2 figure
Multilevel convergence analysis of multigrid-reduction-in-time
This paper presents a multilevel convergence framework for
multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid
estimates. The framework provides a priori upper bounds on the convergence of
MGRIT V- and F-cycles, with different relaxation schemes, by deriving the
respective residual and error propagation operators. The residual and error
operators are functions of the time stepping operator, analyzed directly and
bounded in norm, both numerically and analytically. We present various upper
bounds of different computational cost and varying sharpness. These upper
bounds are complemented by proposing analytic formulae for the approximate
convergence factor of V-cycle algorithms that take the number of fine grid time
points, the temporal coarsening factors, and the eigenvalues of the time
stepping operator as parameters.
The paper concludes with supporting numerical investigations of parabolic
(anisotropic diffusion) and hyperbolic (wave equation) model problems. We
assess the sharpness of the bounds and the quality of the approximate
convergence factors. Observations from these numerical investigations
demonstrate the value of the proposed multilevel convergence framework for
estimating MGRIT convergence a priori and for the design of a convergent
algorithm. We further highlight that observations in the literature are
captured by the theory, including that two-level Parareal and multilevel MGRIT
with F-relaxation do not yield scalable algorithms and the benefit of a
stronger relaxation scheme. An important observation is that with increasing
numbers of levels MGRIT convergence deteriorates for the hyperbolic model
problem, while constant convergence factors can be achieved for the diffusion
equation. The theory also indicates that L-stable Runge-Kutta schemes are more
amendable to multilevel parallel-in-time integration with MGRIT than A-stable
Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material
On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems
Interconnection and damping assignment passivity-based control (IDA-PBC) is
an excellent method to stabilize mechanical systems in the Hamiltonian
formalism. In this paper, several improvements are made on the IDA-PBC method.
The skew-symmetric interconnection submatrix in the conventional form of
IDA-PBC is shown to have some redundancy for systems with the number of degrees
of freedom greater than two, containing unnecessary components that do not
contribute to the dynamics. To completely remove this redundancy, the use of
quadratic gyroscopic forces is proposed in place of the skew-symmetric
interconnection submatrix. Reduction of the number of matching partial
differential equations in IDA-PBC and simplification of the structure of the
matching partial differential equations are achieved by eliminating the
gyroscopic force from the matching partial differential equations. In addition,
easily verifiable criteria are provided for Lyapunov/exponential
stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with
arbitrary degrees of underactuation and for all nonlinear controlled
Hamiltonian systems with one degree of underactuation. A general design
procedure for IDA-PBC is given and illustrated with examples. The duality of
the new IDA-PBC method to the method of controlled Lagrangians is discussed.
This paper renders the IDA-PBC method as powerful as the controlled Lagrangian
method
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
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