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Some New Convergence Analysis and Applications of POD-Greedy Algorithms

Yuwen Li, Yupeng Wang

TL;DR

The paper develops an entropy-number–based convergence theory for the weak POD-Greedy method with multiple POD modes and variable thresholds, connecting the reduced-basis error to the entropy of the solution manifold in the space-time setting. It then extends the framework to an EIM-POD-Greedy scheme that achieves affine-separable approximations of time-dependent parameterized coefficients by combining time-domain POD with space-domain EIM, with convergence analyzed via entropy numbers of the target set. Theoretical results are complemented by numerical experiments showing that multi-mode enrichment can reduce offline cost without sacrificing accuracy and that EIM-POD-Greedy outperforms classical EIM in efficiency, especially when the target set exhibits fast entropy decay. The work provides a practical and rigorous pathway for robust reduced-order modeling of time-dependent parametric PDEs, particularly in cases lacking affine parameter structure.

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.

Some New Convergence Analysis and Applications of POD-Greedy Algorithms

TL;DR

The paper develops an entropy-number–based convergence theory for the weak POD-Greedy method with multiple POD modes and variable thresholds, connecting the reduced-basis error to the entropy of the solution manifold in the space-time setting. It then extends the framework to an EIM-POD-Greedy scheme that achieves affine-separable approximations of time-dependent parameterized coefficients by combining time-domain POD with space-domain EIM, with convergence analyzed via entropy numbers of the target set. Theoretical results are complemented by numerical experiments showing that multi-mode enrichment can reduce offline cost without sacrificing accuracy and that EIM-POD-Greedy outperforms classical EIM in efficiency, especially when the target set exhibits fast entropy decay. The work provides a practical and rigorous pathway for robust reduced-order modeling of time-dependent parametric PDEs, particularly in cases lacking affine parameter structure.

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.
Paper Structure (12 sections, 6 theorems, 78 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 12 sections, 6 theorems, 78 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Weak POD-Greedy Method
  3. Example
  4. POD-Greedy Method
  5. Properties of the POD-Greedy Method
  6. Entropy-Based Convergence Estimate
  7. Entropy Numbers and Convergence Analysis
  8. Width-Based Convergence
  9. EIM-POD-Greedy Method
  10. Numerical Experiments
  11. Weak POD-Greedy Method
  12. EIM-POD-Greedy Method

Key Result

Lemma 2.1

For any $v=(v^0,\ldots,v^J)\in L^2(\mathbb{I};V)$, the eigenvalues of $C_v$ satisfy

Figures (2)

  • Figure 5.1.1: Convergence of $E_N$ with $m$ POD modes against the dimension $N$ of the reduced space (left); convergence history of $\Delta_n$ and $e_n=\|u_{\tau,h,\mu_n}-u_{\tau,n,\mu_n}\|$ with $m=1$ against the number of POD-Greedy iterations (right).
  • Figure 5.2.1: Convergence of the EIM-POD-Greedy method with $m=1,2,3,4$ POD modes against the dimension of the reduced subspace (left); convergence of the EIM and the EIM-POD-Greedy method with $m=1$ against the number of EIM-POD-Greedy iterations (right).

Theorems & Definitions (11)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Theorem 4.1
  • ...and 1 more