On the connection between balanced proper orthogonal decomposition, balanced truncation, and metric complexity theory for infinite dimensional systems
On the connection between balanced proper orthogonal decomposition, balanced truncation, and metric complexity theory for infinite dimensional systems
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无限维系统平衡真正交分解、平衡截断与度量复杂性理论之间的联系
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发表时间：
2010
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通讯作者：
S. Djouadi
中科院分区：
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作者：
S. Djouadi
本文研究了两种重要的模型降阶技术，即平衡本征正交分解(POD)和平衡截断技术之间的关系。特别地，在Hilbert-Schmidt范数下，证明了平衡POD在积分算子空间的距离最小化意义下是最优的。而对于冲激响应满足某些有限能量约束的无限维系统，平衡截断是平衡POD的特例。POD和平衡截断与度量复杂性理论的某些概念相关。特别地，这两种方法都可以最小化偏微分方程解的不同n宽度，包括Kolmogorov解、Gelfand解、线性解和Bernstein解。N宽度量化了由于缺乏数据和信息丢失而导致的固有错误和表示错误。
In this paper, the connection between two important model reduction techniques, namely balanced proper orthogonal decomposition (POD) and balanced truncation is investigated for infinite dimensional systems. In particular, balanced POD is shown to be optimal in the sense of distance minimization in a space of integral operators under the Hilbert-Schmidt norm. Whereas balanced truncation is shown to be a particular case of balanced POD for infinite dimensional systems for which the impulse response satisfies certain finite energy constraints. POD and balanced truncation are related to certain notions of metric complexity theory. In particular both are shown to minimize different n-widths of partial differential equation solutions including the Kolmogorov, Gelfand, linear and Bernstein n-widths. The n-widths quantify inherent and representation errors due to lack of data and loss of information.