  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.
