Abstract:We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an “almost symplectic” part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.

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中文翻译：

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泊松系统的保结构减基方法

摘要：我们通过将其表示为哈密顿动力学系统，开发了针对一类非耗散问题的保留结构的简化基础方法。从这个角度来看，相空间自然地具有泊松流形结构，该结构对动力学的物理性质，对称性和守恒定律进行编码。目标是基于两步法，为一般的与状态有关的退化Poisson结构设计简化的基础方法。首先，通过泊松张量的局部逼近，我们将哈密顿动力学分解为“几乎辛”部分和卡西米尔不变量的琐碎演变。其次，将典型辛减基技术应用到动力学的非平凡分量上，从而精确地保留了局部泊松张量核。在恒定值情况下，简化模型保留了整体泊松结构和相流的守恒性质，在状态依赖情况下，泊松张量逼近误差最大。建立简化系统解的先验误差估计。提出了一组数值模拟，以证实理论上的发现。

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