Rank-adaptive structure-preserving model order reduction of Hamiltonian systems
Rank-adaptive structure-preserving model order reduction of Hamiltonian systems
复制标题
哈密顿系统的秩自适应结构保持模型降阶
DOI：
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复制
发表时间：
2020
期刊：
影响因子：
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通讯作者：
N. Ripamonti
中科院分区：
文献类型：
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作者：
J. Hesthaven;C. Pagliantini;N. Ripamonti
针对有限维参数化哈密顿系统的非耗散现象，提出了一种自适应保结构模型降阶方法。为了克服缓慢衰减的Kolmogorov宽度典型的运输问题，完整的模型近似的本地减少空间，适应在时间上使用动态低秩近似技术。约化动力学通过将切空间中的哈密顿向量场的辛投影近似到局部约化空间来规定。这确保了在约化过程中保持哈密顿动力学的正则辛结构。此外，精确的近似与低秩约化的解决方案，通过允许减少空间的尺寸在时间演化过程中的变化。每当减少的解决方案的质量，通过误差指标评估，是不令人满意的，减少的基础上增加的参数方向是最差的近似由当前的基础。广泛的数值试验，涉及波的相互作用，非线性输运问题，和Vlasov方程证明了优越的上级稳定性和相当大的运行时加速比所提出的方法相比，全球和传统的减少基础的方法。
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.
DOI：
10.1137/17m1140571
发表时间：
2015-12
期刊：
SIAM J. Sci. Comput.
影响因子：
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作者：
J. Reiss;P. Schulze;J. Sesterhenn;V. Mehrmann
通讯作者：
J. Reiss;P. Schulze;J. Sesterhenn;V. Mehrmann