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POD-ROM methods: error analysis for continuous parametrized approximations

Bosco García-Arcilla, Alicia García-Mascaraque, Julia Novo

TL;DR

This work tackles efficient, accurate reduced-order modeling for parametric time-dependent PDEs by introducing a POD-ROM built from finite-difference–style differences in time and parameters. The authors derive a priori, pointwise-in-time error bounds that depend on the tail of the POD spectrum $\\Sigma_r^2=\\sum_{k>r}\lambda_k$ and on discretization parameters $\\Delta t$ and $\\Delta \alpha$, and extend the analysis to two parameters with appropriate higher-order quotients. They also compare the new method to the standard POD approach, showing generally improved accuracy for representative tests and enabling out-of-sample parameter predictions with high fidelity (e.g., periodic orbit periods with relative error around $10^{-9}$). The numerical experiments demonstrate practical viability and guide snapshot design, while the theory provides rigorous guarantees for parametric, time-dependent simulations.

Abstract

This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced order models for parametric equations, a complete error analysis of the methods is still a challenge. We introduce and analyze in this paper a new POD method based on finite differences (respect to time and respect to the parameters that may be considered). We obtain a priori bounds for the new method valid for any value of time in a given time interval and any value of the parameter in a given parameter interval. Our design of the new POD method allow us to prove pointwise-in-time error estimates as opposed to average error bounds obtained typically in POD methods. Most of the papers concerning POD methods for parametric equations are just based on the snapshots computed at different times and parameter values instead of their difference quotients. We show that the error analysis of the present paper can also cover the error analysis of that case (that we call standard). Some numerical experiments compare our new approach with the standard one and support the error analysis.

POD-ROM methods: error analysis for continuous parametrized approximations

TL;DR

This work tackles efficient, accurate reduced-order modeling for parametric time-dependent PDEs by introducing a POD-ROM built from finite-difference–style differences in time and parameters. The authors derive a priori, pointwise-in-time error bounds that depend on the tail of the POD spectrum and on discretization parameters and , and extend the analysis to two parameters with appropriate higher-order quotients. They also compare the new method to the standard POD approach, showing generally improved accuracy for representative tests and enabling out-of-sample parameter predictions with high fidelity (e.g., periodic orbit periods with relative error around ). The numerical experiments demonstrate practical viability and guide snapshot design, while the theory provides rigorous guarantees for parametric, time-dependent simulations.

Abstract

This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced order models for parametric equations, a complete error analysis of the methods is still a challenge. We introduce and analyze in this paper a new POD method based on finite differences (respect to time and respect to the parameters that may be considered). We obtain a priori bounds for the new method valid for any value of time in a given time interval and any value of the parameter in a given parameter interval. Our design of the new POD method allow us to prove pointwise-in-time error estimates as opposed to average error bounds obtained typically in POD methods. Most of the papers concerning POD methods for parametric equations are just based on the snapshots computed at different times and parameter values instead of their difference quotients. We show that the error analysis of the present paper can also cover the error analysis of that case (that we call standard). Some numerical experiments compare our new approach with the standard one and support the error analysis.

Paper Structure

This paper contains 14 sections, 13 theorems, 107 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Proper orthogonal decomposition
  4. Error analysis of the method
  5. The case $\alpha=\alpha_l$
  6. Bound for $\int_0^t \|(I-P^r)u^{\alpha_l}_{h,s}(s)\|_0^2\ \mathrm d s$
  7. Bound for $\int_0^t \|(I-P^r)u^{\alpha_l}_{h}(s)\|_0^2\ \mathrm d s$
  8. The case $\alpha\neq\alpha_l$
  9. Bound for $\int_0^t \|(I-P^r)u^{\alpha}_{h,s}(s)\|_0^2\ \mathrm d s$
  10. Bound for $\int_0^t \|(I-P^r)u^{\alpha}_{h}(s)\|_0^2\ \mathrm d s$
  11. The standard method
  12. Continuous-in-time case with two parameters
  13. Numerical Experiments
  14. Conclusions and future work

Key Result

Lemma 1

Le $X$ be a Banach space and let $z(t_j,\alpha_l)\in X$, $0\le j\le M$, $0\le l\le L$, then

Figures (6)

  • Figure 1: Variation of the time derivative over a period for $\nu=0.01$ and $\beta=4.5$
  • Figure 2: Errors $\|\nabla \boldsymbol e_r(t)\|_0$ (continuous blue line), $\|\nabla {\boldsymbol\varepsilon}_r(t)\|_0$ (discontinuous magenta line), of $\|\nabla \boldsymbol e_r(t_n)\|_0$ (red circles) and $\|{\boldsymbol\varepsilon}_r(t_n)\|_0$ (red crosses) for $\beta=4.25$. Top: $M=32$ on period $[0,T^\beta]$ (left) and detail at the end of the period (right); bottom: detail at the end of the period for $M=64$ and $M=128$.
  • Figure 3: Left: Errors $\max_{t}\| \nabla \boldsymbol e_r(t)\|_0$ (blue continuous), $\max_{t}\| \nabla {\boldsymbol\varepsilon}_r(t)\|_0$ (magenta discontinuous), $\max_{t}\|\nabla( \boldsymbol u_h(t)-\boldsymbol u_{h/2} (t))\|_0$ (black discontinuous) and $\max_{t}\| \nabla \boldsymbol e_r(t)\|_0$ (green continuous) for the traditional POD method; circles mark the values of $\beta$ used for the dataset. Right: Errors $\max_{t}\| \boldsymbol e_r(t)\|_0$ (blue continuous), $\max_{t}\|{\boldsymbol\varepsilon}_r(t)\|_0$ (magenta discontinuous), $\max_{t}\| \boldsymbol u_h(t)-\boldsymbol u_{h/2} (t)\|_0$ (black discontinuous) and $\max_{t}\| \boldsymbol e_r(t)\|_0$ (green continuous) for the traditional POD method.
  • Figure 4: Two parameters: accuracy of the FEM approximations used in the data set
  • Figure 5: Two parameters: Left, $\max_{t\in [0,T^{\nu_l,\beta_k}]}\|\nabla \boldsymbol e_r(t)\|_0$ (continuous line) and $\max_{t\in [0,T^{\nu_l,\beta_k}]}\| \nabla {\boldsymbol\varepsilon}_r(t)\|_0$ (discontinuous line); Right, ratios $\max_{t\in [0,T^{\nu_l,\beta_k}]}\| \nabla \boldsymbol e_r(t)\|_0/\max_{t\in [0,T^{\nu_l,\beta_k}]}\| \nabla {\boldsymbol\varepsilon}_r(t)\|_0$.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Proof 3
  • Lemma 5
  • Proof 4
  • ...and 12 more