A new analysis of empirical interpolation methods and Chebyshev greedy algorithms

Summary

The paper presents new convergence estimates for the Empirical Interpolation Method (EIM) and the Chebyshev Greedy Algorithm (CGA) in terms of entropy numbers, providing sharper convergence rates than classical analysis.

Highlights

• The EIM is a popular tool for reduced-order modeling and analysis of complex nonlinear systems.
• The CGA is a natural generalization of the Orthogonal Greedy Algorithm (OGA) in Banach spaces.
• Entropy numbers are used to describe the massiveness of the symmetric convex hull of the underlying function set.
• The paper derives novel convergence estimates for the EIM and CGA in terms of entropy numbers.
• The estimates are sharper than classical results and hold without assuming the monotonicity of the Lebesgue constant.
• The paper also presents numerical experiments to illustrate the convergence rates of the EIM and CGA.
• The results have implications for the efficiency of reduced-order modeling and sparse approximation.

Key Insights

• The EIM and CGA are widely used algorithms in reduced-order modeling and sparse approximation, and their convergence rates are crucial for understanding their efficiency.
• The use of entropy numbers provides a more precise description of the complexity of the underlying function set, leading to sharper convergence estimates.
• The paper's results highlight the importance of considering the Banach-Mazur distance between the underlying Banach space and the Euclidean space.
• The numerical experiments demonstrate the influence of the regularity index and the type of Banach space on the convergence rates of the EIM and CGA.
• The paper's findings have implications for the development of new algorithms and the improvement of existing ones in reduced-order modeling and sparse approximation.
• The use of entropy numbers and Banach-Mazur distances provides a new perspective on the analysis of greedy algorithms in Banach spaces.
• The results of the paper can be extended to other greedy algorithms and have the potential to lead to new breakthroughs in the field.

Mindmap

Citation

Li, Y. (2024). A new analysis of empirical interpolation methods and Chebyshev greedy algorithms (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2401.13985