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To introduce the Proper Orthogonal Decomposition, POD, let us begin by assuming that some solutions of the unknown field of interest \(u(\textbf{x},t)\) are known at particular nodal positions \(\textbf{x}_i\) at discrete time instants \(t_m=m \Delta t\) , with \(i\in [1,\ldots , \texttt{N}_n]\) and \(m \in [1,\ldots , M]\) . From now on, we denote \(u(\textbf{x}_i,t_m) \equiv u^m(\textbf{x}_i)\equiv u^m_i\) and define \(\textbf{u}^m\) as the vector of nodal values \(u^m_i\) at time \(t_m\) .

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Proper Orthogonal Decomposition and Reduced Basis

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  • 出版模式：

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  • 发表日期：

    2025年

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  • Francisco Chinesta,
  • Elías Cueto,
  • Victor Champaney,
  • Chady Ghnatios,
  • Amine Ammar,
  • Nicolas Hascoët,
  • David González,
  • Icíar Alfaro,
  • Daniele Di Lorenzo,
  • Angelo Pasquale,
  • Dominique Baillargeat

摘要

To introduce the Proper Orthogonal Decomposition, POD, let us begin by assuming that some solutions of the unknown field of interest \(u(\textbf{x},t)\) are known at particular nodal positions \(\textbf{x}_i\) at discrete time instants \(t_m=m \Delta t\) , with \(i\in [1,\ldots , \texttt{N}_n]\) and \(m \in [1,\ldots , M]\) . From now on, we denote \(u(\textbf{x}_i,t_m) \equiv u^m(\textbf{x}_i)\equiv u^m_i\) and define \(\textbf{u}^m\) as the vector of nodal values \(u^m_i\) at time \(t_m\) .