9,489 research outputs found
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure
Structure Preserving Model Reduction of Parametric Hamiltonian Systems
While reduced-order models (ROMs) have been popular for efficiently solving
large systems of differential equations, the stability of reduced models over
long-time integration is of present challenges. We present a greedy approach
for ROM generation of parametric Hamiltonian systems that captures the
symplectic structure of Hamiltonian systems to ensure stability of the reduced
model. Through the greedy selection of basis vectors, two new vectors are added
at each iteration to the linear vector space to increase the accuracy of the
reduced basis. We use the error in the Hamiltonian due to model reduction as an
error indicator to search the parameter space and identify the next best basis
vectors. Under natural assumptions on the set of all solutions of the
Hamiltonian system under variation of the parameters, we show that the greedy
algorithm converges with exponential rate. Moreover, we demonstrate that
combining the greedy basis with the discrete empirical interpolation method
also preserves the symplectic structure. This enables the reduction of the
computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy,
and stability of this model reduction technique is illustrated through
simulations of the parametric wave equation and the parametric Schrodinger
equation
Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds
This work presents two novel approaches for the symplectic model reduction of
high-dimensional Hamiltonian systems using data-driven quadratic manifolds.
Classical symplectic model reduction approaches employ linear symplectic
subspaces for representing the high-dimensional system states in a
reduced-dimensional coordinate system. While these approximations respect the
symplectic nature of Hamiltonian systems, linear basis approximations can
suffer from slowly decaying Kolmogorov -width, especially in wave-type
problems, which then requires a large basis size. We propose two different
model reduction methods based on recently developed quadratic manifolds, each
presenting its own advantages and limitations. The addition of quadratic terms
to the state approximation, which sits at the heart of the proposed
methodologies, enables us to better represent intrinsic low-dimensionality in
the problem at hand. Both approaches are effective for issuing predictions in
settings well outside the range of their training data while providing more
accurate solutions than the linear symplectic reduced-order models
Symplectic Model-Reduction with a Weighted Inner Product
In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm bounded reduced basis. The accuracy and the stability of this model reduction technique is illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation
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
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