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The non-intrusive reduced basis two-grid method applied to sensitivity analysis

Elise Grosjean, Bernd Simeon

TL;DR

The paper develops a non-intrusive reduced basis two-grid framework for sensitivity analysis of parabolic PDEs, addressing both direct and adjoint methods. By combining offline RB generation on a fine mesh with online solves on a coarse mesh, and augmenting with time interpolation and rectification, the method achieves near-optimal convergence in $L^{ abla}(0,T;H^1_0(Ω))$ for sensitivities while substantially reducing online cost. A key contribution is the explicit NIRB error bound for sensitivities, plus a Gaussian Process Regression-based online enhancement that predicts projection coefficients to further accelerate computations. The numerical experiments on the heat equation and Brusselator demonstrate accurate sensitivity approximations with significant speedups, validating the method’s practicality for complex parametric problems. Overall, the work offers a scalable, non-intrusive pathway to efficient, accurate sensitivity analysis in parametric parabolic systems with potential applications in optimization and uncertainty quantification.

Abstract

This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly. To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue, non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a ``black-box'', requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation. In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in $L^{\infty}(0,T;H^1(Ω))$, and then establish that these rates are recovered by the proposed NIRB approximation. These results are supported by numerical simulations. We then numerically demonstrate that a Gaussian process regression can be used to approximate the projection coefficients of the NIRB two-grid method. This further reduces the computational costs of the online step while only computing a coarse solution of the initial problem. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.

The non-intrusive reduced basis two-grid method applied to sensitivity analysis

TL;DR

The paper develops a non-intrusive reduced basis two-grid framework for sensitivity analysis of parabolic PDEs, addressing both direct and adjoint methods. By combining offline RB generation on a fine mesh with online solves on a coarse mesh, and augmenting with time interpolation and rectification, the method achieves near-optimal convergence in for sensitivities while substantially reducing online cost. A key contribution is the explicit NIRB error bound for sensitivities, plus a Gaussian Process Regression-based online enhancement that predicts projection coefficients to further accelerate computations. The numerical experiments on the heat equation and Brusselator demonstrate accurate sensitivity approximations with significant speedups, validating the method’s practicality for complex parametric problems. Overall, the work offers a scalable, non-intrusive pathway to efficient, accurate sensitivity analysis in parametric parabolic systems with potential applications in optimization and uncertainty quantification.

Abstract

This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly. To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue, non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a ``black-box'', requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation. In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in , and then establish that these rates are recovered by the proposed NIRB approximation. These results are supported by numerical simulations. We then numerically demonstrate that a Gaussian process regression can be used to approximate the projection coefficients of the NIRB two-grid method. This further reduces the computational costs of the online step while only computing a coarse solution of the initial problem. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.
Paper Structure (24 sections, 5 theorems, 32 equations, 4 tables)

This paper contains 24 sections, 5 theorems, 32 equations, 4 tables.

Table of Contents

  1. Introduction.
  2. Mathematical Background.
  3. A model problem: The heat equation.
  4. The continuous problem.
  5. The various discretizations.
  6. Sensitivity analysis: The direct problem.
  7. The continuous setting.
  8. The spatially semi-discretized version.
  9. The fully-discretized versions.
  10. Sensitivity analysis: The adjoint problem.
  11. The continuous setting.
  12. Discretized setting.
  13. The Non-Intrusive Reduced Basis method (NIRB) in the context of parabolic equations.
  14. NIRB error estimate on the sensitivities
  15. Main result
  16. ...and 9 more sections

Key Result

Theorem 2.2

Let $\Omega$ be a convex polyhedron. Let $A(\mu)=\mu \ I_d$, with $\mu \in \mathbb{R}_*^+$. Consider $u \in H^{1}(0,T;\ H^2(\Omega)) \cap H^{2}(0,T;\ L^2(\Omega))$ to be the solution of heatEq2 with $u^0 \in H^2(\Omega)$ and $u_h^n$ be the fully-discretized variational form varpara2disc. Let $\Psi$

Theorems & Definitions (12)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6: of Theorem \ref{['corollaryPsiiCrankNicolson']}
  • Theorem 3.1
  • proof
  • ...and 2 more