  Mathematical modeling and simulation of complex physical systems based on
partial differential equations (PDEs) have been widely used in engineering and
industrial applications. To enable reliable predictions, it is crucial yet
challenging to calibrate the model by inferring unknown parameters/fields
(e.g., boundary conditions, mechanical properties, and operating parameters)
from sparse and noisy measurements, which is known as a PDE-constrained inverse
problem. In this work, we develop a novel bi-fidelity (BF) ensemble Kalman
inversion method to tackle this challenge, leveraging the accuracy of
high-fidelity models and the efficiency of low-fidelity models. The core
concept is to build a BF model with a limited number of high-fidelity samples
for efficient forward propagations in the iterative ensemble Kalman inversion.
Compared to existing inversion techniques, salient features of the proposed
methods can be summarized as follow: (1) achieving the accuracy of
high-fidelity models but at the cost of low-fidelity models, (2) being robust
and derivative-free, and (3) being code non-intrusive, enabling ease of
deployment for different applications. The proposed method has been assessed by
three inverse problems that are relevant to fluid dynamics, including both
parameter estimation and field inversion. The numerical results demonstrate the
excellent performance of the proposed BF ensemble Kalman inversion approach,
which drastically outperforms the standard Kalman inversion in terms of
efficiency and accuracy.
