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Structure preserving model reduction of parametric Hamiltonian systems. (English) Zbl 1379.78019

This paper deals with a greedy approach for the construction of a reduced system that preserves the geometric structure of Hamiltonian systems. The authors are interested in finding the best basis vectors that increase the overall accuracy of the reduced basis. In the case of compact subsets with exponentially small Kolmogorov \(n\)-width, the fast convergence of the greedy algorithm is recovered exponentially. In the framework of the fast approximation of nonlinear terms, the basis obtained by the greedy method is combined with a discrete empirical interpolation method (resp., symplectic discrete empirical interpolation method) in order to construct a reduced system with a Hamiltonian that is arbitrarily close to the Hamiltonian of the original system. The last section of this paper is concerned with the numerical convergence of the symplectic greedy method. The numerical results demonstrate that the greedy method can save substantial computational cost in the offline stage as compared to alternative singular value decomposition-based techniques.

MSC:

78M34 Model reduction in optics and electromagnetic theory
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
78A40 Waves and radiation in optics and electromagnetic theory

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