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.

翻译：本文研究了用于灵敏度分析的非侵入式降阶基（NIRB）技术推导，具体涉及直接法和伴随状态法。对于高度复杂的参数问题，这两种方法可能变得过于昂贵。为降低计算时间，已有研究采用本征正交分解（POD）和降阶基方法（RBMs）。然而，这些算法大多具有侵入性，即需要修改高保真（HF）代码。为解决此问题，本文采用非侵入式策略。NIRB双网格方法仅将HF代码作为“黑箱”使用，无需修改代码。与其他RBM类似，该方法基于离线-在线分解。离线阶段耗时但仅执行一次，而在线阶段的计算成本远低于HF评估。本文针对直接法和伴随状态法提出了新的NIRB双网格算法。在直接法中，我们以经典模型问题——热方程为例，证明了灵敏度的高保真评估在$L^{\infty}(0,T;H^1(\Omega))$中达到最优收敛速率，并进一步证明所提出的NIRB近似能够恢复这些收敛速率。数值模拟结果支持了上述结论。随后，我们通过数值实验表明，高斯过程回归可用于近似NIRB双网格方法的投影系数。这进一步降低了在线步骤的计算成本，同时仅需计算初始问题的粗网格解。所有数值结果均基于模型问题及更复杂的Brusselator系统进行测试。