  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.
