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

TL;DR

This paper introduces non-intrusive reduced basis two-grid algorithms for sensitivity analysis that significantly reduce computational costs while maintaining accuracy, applicable to complex parametric problems like the heat equation and Brusselator system.

Contribution

It develops new NIRB two-grid algorithms for direct and adjoint sensitivity methods that do not require code modification and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Findings

NIRB two-grid method achieves optimal convergence rates for sensitivities.

Gaussian process regression further reduces online computational costs.

Numerical tests confirm the method's efficiency on heat equation and Brusselator system.

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 $L^{\infty}(0,T;H^1(\Omega))$, 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.