3,611 research outputs found
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems
Reduced basis methods are popular for approximately solving large and complex
systems of differential equations. However, conventional reduced basis methods
do not generally preserve conservation laws and symmetries of the full order
model. Here, we present an approach for reduced model construction, that
preserves the symplectic symmetry of dissipative Hamiltonian systems. The
method constructs a closed reduced Hamiltonian system by coupling the full
model with a canonical heat bath. This allows the reduced system to be
integrated with a symplectic integrator, resulting in a correct dissipation of
energy, preservation of the total energy and, ultimately, in the stability of
the solution. Accuracy and stability of the method are illustrated through the
numerical simulation of the dissipative wave equation and a port-Hamiltonian
model of an electric circuit
Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems
This work proposes an adaptive structure-preserving model order reduction
method for finite-dimensional parametrized Hamiltonian systems modeling
non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width
typical of transport problems, the full model is approximated on local reduced
spaces that are adapted in time using dynamical low-rank approximation
techniques. The reduced dynamics is prescribed by approximating the symplectic
projection of the Hamiltonian vector field in the tangent space to the local
reduced space. This ensures that the canonical symplectic structure of the
Hamiltonian dynamics is preserved during the reduction. In addition, accurate
approximations with low-rank reduced solutions are obtained by allowing the
dimension of the reduced space to change during the time evolution. Whenever
the quality of the reduced solution, assessed via an error indicator, is not
satisfactory, the reduced basis is augmented in the parameter direction that is
worst approximated by the current basis. Extensive numerical tests involving
wave interactions, nonlinear transport problems, and the Vlasov equation
demonstrate the superior stability properties and considerable runtime speedups
of the proposed method as compared to global and traditional reduced basis
approaches
Energy-Storage Balanced Reduction of Port-Hamiltonian Systems
Supported by the framework of dissipativity theory, a procedure based on physical energy to balance and reduce port-Hamiltonian systems with collocated inputs and outputs is presented. Additionally, some relations with the methods of nonlinear balanced reduction are exposed. Finally a structure-preserving reduction method based on singular perturbations is shown.
An Overview of Variational Integrators
The purpose of this paper is to survey some recent advances in variational
integrators for both finite dimensional mechanical systems as well as continuum
mechanics. These advances include the general development of discrete
mechanics, applications to dissipative systems, collisions, spacetime integration algorithms,
AVI’s (Asynchronous Variational Integrators), as well as reduction for
discrete mechanical systems. To keep the article within the set limits, we will only
treat each topic briefly and will not attempt to develop any particular topic in
any depth. We hope, nonetheless, that this paper serves as a useful guide to the
literature as well as to future directions and open problems in the subject
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
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