Some New Convergence Analysis and Applications of POD-Greedy Algorithms

TL;DR

This paper introduces new convergence estimates for POD-Greedy algorithms with multiple modes and thresholds, and combines POD with EIM for improved parametric function approximation, supported by numerical experiments.

Contribution

It provides novel convergence analysis for POD-Greedy methods and proposes an EIM-POD-Greedy approach with entropy-based guarantees.

Findings

New convergence estimates for weak POD-Greedy methods.

Effective EIM-POD-Greedy algorithm demonstrated.

Numerical results show improved performance over traditional methods.

Abstract

In this article, we derive a novel convergence estimate for the weak POD-Greedy method with multiple POD modes and variable greedy thresholds in terms of the entropy numbers of the parametric solution manifold. Combining the POD with the Empirical Interpolation Method (EIM), we also propose an EIM-POD-Greedy method with entropy-based convergence analysis for simultaneously approximating parametrized target functions by separable approximants. Several numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm compared to traditional methods.