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Convergence analysis for a tree-based nonlinear reduced basis method

Mohamed Barakat, Diane Guignard

TL;DR

The paper addresses efficient reduced-order modeling for parametrized elliptic PDEs by introducing a nonlinear RB method that partitions the parameter domain into tensor-product subdomains via a binary-tree, assigning a local RB space to each leaf. It proves convergence bounds for arbitrary parameter dimension $d$ and RB size $N$, showing the number of subdomains scales as $K(\\epsilon) = O(\\epsilon^{-d/N^{1/d}})$ under quasi-uniformity and holomorphic parameter dependence. The approach yields low online storage due to tensor-product geometry and demonstrates favorable convergence in diffusion and convection–diffusion tests, often outperforming existing nonlinear RB methods. The work also provides a model problem to illustrate the theoretical assumptions and offers directions for extending to time-dependent and nonlinear PDEs with adaptive partitioning.

Abstract

We develop and analyze a nonlinear reduced basis (RB) method for parametrized elliptic partial differential equations based on a binary-tree partition of the parameter domain into tensor-product structured subdomains. Each subdomain is associated with a local RB space of prescribed dimension, constructed via a greedy algorithm. A splitting strategy along the longest edge of the parameter subdomains ensures geometric control of the subdomains and enables a rigorous convergence analysis. Under the assumption that the parameter-to-solution map admits a holomorphic extension and that the resulting domain partition is quasi-uniform, we establish explicit bounds on the number of subdomains required to achieve a given tolerance for arbitrary parameter domain dimension and RB spaces size. Numerical experiments for diffusion and convection-diffusion problems confirm the theoretical predictions, demonstrating that the proposed approach, which has low storage requirements, achieves the expected convergence rates and in several cases outperforms an existing nonlinear RB method.

Convergence analysis for a tree-based nonlinear reduced basis method

TL;DR

The paper addresses efficient reduced-order modeling for parametrized elliptic PDEs by introducing a nonlinear RB method that partitions the parameter domain into tensor-product subdomains via a binary-tree, assigning a local RB space to each leaf. It proves convergence bounds for arbitrary parameter dimension and RB size , showing the number of subdomains scales as under quasi-uniformity and holomorphic parameter dependence. The approach yields low online storage due to tensor-product geometry and demonstrates favorable convergence in diffusion and convection–diffusion tests, often outperforming existing nonlinear RB methods. The work also provides a model problem to illustrate the theoretical assumptions and offers directions for extending to time-dependent and nonlinear PDEs with adaptive partitioning.

Abstract

We develop and analyze a nonlinear reduced basis (RB) method for parametrized elliptic partial differential equations based on a binary-tree partition of the parameter domain into tensor-product structured subdomains. Each subdomain is associated with a local RB space of prescribed dimension, constructed via a greedy algorithm. A splitting strategy along the longest edge of the parameter subdomains ensures geometric control of the subdomains and enables a rigorous convergence analysis. Under the assumption that the parameter-to-solution map admits a holomorphic extension and that the resulting domain partition is quasi-uniform, we establish explicit bounds on the number of subdomains required to achieve a given tolerance for arbitrary parameter domain dimension and RB spaces size. Numerical experiments for diffusion and convection-diffusion problems confirm the theoretical predictions, demonstrating that the proposed approach, which has low storage requirements, achieves the expected convergence rates and in several cases outperforms an existing nonlinear RB method.

Paper Structure

This paper contains 14 sections, 4 theorems, 121 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Linear reduced basis method
  4. Greedy algorithm
  5. Error estimator
  6. Nonlinear reduced basis method
  7. Tree-based structure for the subdomains
  8. The approximation algorithm
  9. Convergence analysis
  10. Convergence of $N$-term truncated polynomial expansion
  11. Convergence theory of the nonlinear RB method
  12. A model problem
  13. Numerical results
  14. Conclusion

Key Result

Theorem 3.1

Let $F$ be an arbitrary compact set in a Hilbert space $V$. For each $N = 1, 2,...,$ we have where $\kappa > 2$ is a constant depending on the effectivity of the error estimator.

Figures (10)

  • Figure 1: Solutions of Problem \ref{['fe model']} for different parameter values: (a) $\mu=(-1,1)$, and (b) $\mu=(1,1)$.
  • Figure 2: (a) Greedy parameter selection. (b)Linear RB convergence results.
  • Figure 3: Convergence results for the two algorithms at $N=1,2,$ and $4$.
  • Figure 4: Final parameter domain partition when $N=4$ and $\epsilon=5\times10^{-5}$ for (a) the proposed algorithm (b) the algorithm from eftang2010hp.
  • Figure 5: Solutions of Problem \ref{['FE prob']} for different parameter values: (a) $\mu=(0,10)$, (b) $\mu=(0,0)$ and (c) $\mu=(\pi,10)$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma 3.1
  • Remark 4.1
  • Remark 4.2
  • Proposition 4.1
  • proof
  • Theorem 4.1
  • proof
  • Example 5.1
  • Example 5.2