Scientific and engineering problems often involve parametric partial differential equations (PDEs), such as uncertainty quantification, optimizations, and inverse problems. However, solving these PDEs repeatedly can be prohibitively expensive, especially for large-scale complex applications. To address this issue, reduced order modeling (ROM) has emerged as an effective method to reduce computational costs. However, ROM often requires significant modifications to the existing code, which can be time-consuming and complex, particularly for large-scale legacy codes. Non-intrusive methods have gained attention as an alternative approach. However, most existing non-intrusive approaches are purely data-driven and may not respect the underlying physics laws during the online stage, resulting in less accurate approximations of the reduced solution. In this study, we propose a new non-intrusive bi-fidelity reduced basis method for time-independent parametric PDEs. Our algorithm utilizes the discrete operator, solutions, and right-hand sides obtained from the high-fidelity legacy solver. By leveraging a low-fidelity model, we efficiently construct the reduced operator and right-hand side for new parameter values during the online stage. Unlike other non-intrusive ROM methods, we enforce the reduced equation during the online stage. In addition, the non-intrusive nature of our algorithm makes it straightforward and applicable to general nonlinear time-independent problems. We demonstrate its performance through several benchmark examples, including nonlinear and multiscale PDEs.

翻译：科学与工程问题常涉及参数化偏微分方程（PDEs），例如不确定性量化、优化和反问题。然而，重复求解这些PDEs的计算成本可能极为高昂，尤其对于大规模复杂应用。为应对这一问题，降阶建模（ROM）作为降低计算成本的有效方法应运而生。然而，ROM通常需要对现有代码进行大量修改，这可能耗时且复杂，尤其对于大规模遗留代码。非侵入式方法作为一种替代方案受到关注。然而，现有大多数非侵入式方法纯粹基于数据驱动，在线阶段可能不遵循底层物理定律，导致降阶解的近似精度较低。本研究针对时间无关的参数化PDEs提出了一种新的非侵入式双保真度降阶基方法。我们的算法利用从高保真度遗留求解器中获得的离散算子、解及右端项。通过借助低保真度模型，我们在在线阶段高效地构建新参数值下的降阶算子和右端项。与其他非侵入式ROM方法不同，我们在在线阶段强制执行降阶方程。此外，该算法的非侵入式特性使其易于推广至一般非线性时间无关问题。我们通过多个基准算例（包括非线性和多尺度PDEs）验证了其性能。