Space-time POD and the Hankel matrix

Area of Science:

• Dynamical Systems and Control Theory
• Data-Driven Modeling
• Scientific Computing

Background:

• Time-delay embedding is a foundational technique for data-driven reduced-order modeling.
• Singular value decomposition (SVD) of block Hankel matrices is central to popular reduced-order modeling methods.
• Understanding the theoretical underpinnings of Hankel modes is crucial for advancing these methods.

Purpose of the Study:

• To establish a theoretical connection between Hankel modes derived from time-delay embedding and space-time proper orthogonal decomposition (POD) modes.
• To provide a clear interpretation of Hankel modes by relating them to classical POD theory.
• To identify opportunities for improving the accuracy and computational efficiency of reduced-order models.

Main Methods:

• Formulation of a block Hankel matrix from successive delay embeddings of a dynamical system's state.
• Application of singular value decomposition (SVD) to the block Hankel matrix.
• Analysis of the left singular vectors and singular values in relation to space-time POD modes and energies.
• Investigation using the correlation matrix derived from the Hankel matrix.

Main Results:

• Left singular vectors of the Hankel matrix are discrete approximations of space-time POD modes.
• Singular values correspond to the square roots of POD energies.
• Insights into Hankel mode interpretation, including the meaning of rows/columns, optimal norms, time-step impact, and embedding dimension effects.
• Demonstration that standard space-only POD and spectral POD are limiting cases of this framework.

Conclusions:

• The study provides a rigorous theoretical link between SVD of Hankel matrices and space-time POD.
• This connection enhances the interpretability and theoretical grounding of data-driven reduced-order modeling techniques.
• The established relationships offer pathways to improve computational efficiency and accuracy in practical applications.