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

¹ UMA, ENSTA-Paris, 91120 Palaiseau, France
² Department of Mathematics, RPTU Kaiserslautern-Landau, 67657 Deutschland, Germany

^* Corresponding author: elise.grosjean@ensta-paris.fr

Received: 20 January 2023
Accepted: 31 May 2024

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 ans thus Reduced Basis Methods (RBMs) may be a viable option. We propose new NIRB two-grid algorithms for both the direct and adjoint state methods in the context of parabolic equations. 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 employs coarser grids and thus, is significantly less expensive than a fine HF evaluation. 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^∞(0, T ; H¹₀(Ω)), and then establish that these rates are recovered by the NIRB two-grid approximation. These results are supported by numerical simulations. We then propose a new procedure that further reduces the computational costs of the online step while only computing a coarse solution of the state equations. On the adjoint state method, we propose a new algorithm that reduces both the state and adjoint solutions. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.