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Nonlinear compressive reduced basis approximation : when Taylor meets Kolmogorov

Joubine Aghili, Hassan Ballout, Yvon Maday, Christophe Prud'homme

Abstract

This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vecμ \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough, it is effective to approximate the solution as a sum of $N$ separable terms, each being the product of a parameter-dependent coefficient and a space-dependent function. This leads to reduced-order models with $N$ degrees of freedom and complexity of order ${\mathcal O}(N^3)$. However, when the $N$-width decays slowly, $N$ must be large to achieve acceptable accuracy, making cubic complexity prohibitive. The linear complexity measure in terms of Kolmogorov width must be replaced by the Gelfand width, with its associated sensing number. Recent nonlinear approaches based on this notion decompose the $N$ coordinates into two groups: $n$ free variables and $\overline{n}$ dependent variables, where the latter are nonlinear functions of the former ($N= n+\overline n$). Several works have focused on cases where these $\overline{n}$ functions are homogeneous quadratic forms of the $n$ variables, with optimization strategies for choosing $n$ given a target accuracy. A rigorous analysis of the local sensing number is carried out, showing that $n = p$ is optimal and appropriate, at least locally, around a reference point. In practical scenarios involving wide parameter ranges, the condition $p\le n \le p + k$ (with $k$ small) is valid and more robust from continuity arguments. Additionally, the assumption of a quadratic mapping, while justified in a local sense, becomes insufficient. More expressive nonlinear mappings-including those using machine learning-become necessary. This work contributes a theoretical foundation for such strategies and highlights the need for further investigations to push back the Kolmogorov Barrier.

Nonlinear compressive reduced basis approximation : when Taylor meets Kolmogorov

Abstract

This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector . In cases where the Kolmogorov -width decays fast enough, it is effective to approximate the solution as a sum of separable terms, each being the product of a parameter-dependent coefficient and a space-dependent function. This leads to reduced-order models with degrees of freedom and complexity of order . However, when the -width decays slowly, must be large to achieve acceptable accuracy, making cubic complexity prohibitive. The linear complexity measure in terms of Kolmogorov width must be replaced by the Gelfand width, with its associated sensing number. Recent nonlinear approaches based on this notion decompose the coordinates into two groups: free variables and dependent variables, where the latter are nonlinear functions of the former (). Several works have focused on cases where these functions are homogeneous quadratic forms of the variables, with optimization strategies for choosing given a target accuracy. A rigorous analysis of the local sensing number is carried out, showing that is optimal and appropriate, at least locally, around a reference point. In practical scenarios involving wide parameter ranges, the condition (with small) is valid and more robust from continuity arguments. Additionally, the assumption of a quadratic mapping, while justified in a local sense, becomes insufficient. More expressive nonlinear mappings-including those using machine learning-become necessary. This work contributes a theoretical foundation for such strategies and highlights the need for further investigations to push back the Kolmogorov Barrier.
Paper Structure (41 sections, 11 theorems, 150 equations, 8 figures, 3 algorithms)

This paper contains 41 sections, 11 theorems, 150 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Linear approximation and reduced basis construction
  3. Singular Value Decomposition
  4. Optimality property.
  5. Centered vs. non-centered snapshots.
  6. Greedy Algorithms
  7. GSS: Greedy-Sampled SVD
  8. Advantages over SVD/POD
  9. Advantages over Greedy
  10. Nonlinear approximation
  11. Nonlinear widths
  12. Nonlinear Compressive Reduced Basis Approximation
  13. Approximation with Taylor polynomial expansions
  14. Notations and Assumptions
  15. Tangent space of the centered manifold
  16. ...and 26 more sections

Key Result

Theorem 1

As $r \to 0$, the subspace $\mathcal{U}^r_{1:q}$ spanned by the first $q$ left singular vectors of $\mathbf{S}^r$ converges to the tangent space $T_0^*$ with rate $\mathcal{O}(r)$, i.e.

Figures (8)

  • Figure 1: Comparison of quadratic approximation using homogeneous (in blue marks) and full quadratic (in red marks) feature maps on the toy example of \ref{['sec:problem_formulation']}. The QSVDM and QGM methods are equivalent for this example.
  • Figure 2: Schematic of thermal fin geometry with labeled subdomains $\Omega_i$ and boundaries $\Gamma_{\text{root}}$, $\Gamma_{\text{ext}}$.
  • Figure 3: Comparison of mean and max relative errors for different basis construction algorithms.
  • Figure 4: Convergence of the first $p$ SVD modes of $\mathbf{S}^r$ for dimension $p \in \{1,3,5\} .$
  • Figure 5: Convergence of the first $m$ SVD modes of $\widehat{\mathbf S}^{r} = (\mathbf I-\Pi_0)\mathbf{S}^r$ for dimension $p \in \{1,2,3\}.$
  • ...and 3 more figures

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Theorem 1: Convergence to the tangent space
  • proof
  • Proposition 2: Convergence of the coefficients
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 17 more