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Accelerating High-Fidelity Fixed Point Schemes with On-the-fly Reduced Order Modeling

Philippe-André Luneau, Jean Deteix

TL;DR

This work introduces an on-the-fly reduced-order modeling framework to accelerate fixed-point solvers for PDEs without offline training. An adaptive, a posteriori error estimator governs the ROM accuracy by propagating the error through the iteration, ensuring fidelity to the high-fidelity solution. The authors develop a ROM-based inexact operator for multi-system fixed-point problems, prove convergence under Lipschitz and contraction assumptions, and provide explicit error bounds. Applied to a coupled Stokes–heat problem, the method yields meaningful speedups while preserving accuracy, and demonstrates robustness to parameter variations and basis size choices.

Abstract

A general method for accelerating fixed point schemes for problems related to partial differential equations is presented in this article. The speedup is obtained by training a reduced-order model on-the-fly, removing the need to do an offline training phase and any dependence to a precomputed reduced basis (e.g. a fixed geometry or mesh). The surrogate model can adapt itself along the iterations because of an error criterion based on error propagation, ensuring the high fidelity of the converged result. Convergence results are given for a general class of fixed point problems with complex dependence structures between multiple auxiliary linear systems. The proposed algorithm is applied to the solution of a system of coupled partial differential equations. The speedups obtained are significant, and the output of the method can be considered high-fidelity when compared to the reference solution.

Accelerating High-Fidelity Fixed Point Schemes with On-the-fly Reduced Order Modeling

TL;DR

This work introduces an on-the-fly reduced-order modeling framework to accelerate fixed-point solvers for PDEs without offline training. An adaptive, a posteriori error estimator governs the ROM accuracy by propagating the error through the iteration, ensuring fidelity to the high-fidelity solution. The authors develop a ROM-based inexact operator for multi-system fixed-point problems, prove convergence under Lipschitz and contraction assumptions, and provide explicit error bounds. Applied to a coupled Stokes–heat problem, the method yields meaningful speedups while preserving accuracy, and demonstrates robustness to parameter variations and basis size choices.

Abstract

A general method for accelerating fixed point schemes for problems related to partial differential equations is presented in this article. The speedup is obtained by training a reduced-order model on-the-fly, removing the need to do an offline training phase and any dependence to a precomputed reduced basis (e.g. a fixed geometry or mesh). The surrogate model can adapt itself along the iterations because of an error criterion based on error propagation, ensuring the high fidelity of the converged result. Convergence results are given for a general class of fixed point problems with complex dependence structures between multiple auxiliary linear systems. The proposed algorithm is applied to the solution of a system of coupled partial differential equations. The speedups obtained are significant, and the output of the method can be considered high-fidelity when compared to the reference solution.

Paper Structure

This paper contains 16 sections, 8 theorems, 84 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fixed Point Iterations
  4. Projection-based ROMs and Full-field Approximations
  5. Methodology
  6. Error Propagation
  7. ROM-based Inexact Operator
  8. Picard Iterations Depending on the Solutions of Multiple Linear Systems
  9. ROM Approximation
  10. Application
  11. Alternate Solution of Coupled Equations
  12. Constants Estimation
  13. Numerical Results
  14. Comparison of ROM Quality Measures
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

Let $G$, $H$ be $\mathbb{R}^n$-selfmaps with respective Lipschitz constants $L_1$, $L_2$ s.t. $L_1L_2<1$. Assuming the sequences $\mathbf{x}^{k+1} = H(G(\mathbf{x}^k))$ and $\widehat{\mathbf{x}}^{k+1} = H_k(G_k(\widehat{\mathbf{x}}^k))$ both converge to $\mathbf{x}^*$ and $\widehat{\mathbf{x}}^*$ re

Figures (6)

  • Figure 1: The vertical tube geometry.
  • Figure 2: Fields and errors w.r.t. the reference solution for the solution obtained with the U-ROM-accelerated fixed point scheme with $N_b=5$, $\varepsilon_\mathrm{rb}=10^{-7}$.
  • Figure 3: Propagated error along iterations for U-ROM (left) and T-ROM (right), for different maximal basis sizes $N_b$.
  • Figure 4: Propagated error along iterations for U-ROM (left) and T-ROM (right), for different reduced basis tolerance $\varepsilon_\mathrm{rb}$.
  • Figure 5: Speedups for the total time (left) and the FOM solve time (right) for each variants of the algorithm, on 1 and 4 CPUs.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • Theorem 4
  • proof
  • Remark 1
  • ...and 13 more