  Low-fidelity data is typically inexpensive to generate but inaccurate. On the
other hand, high-fidelity data is accurate but expensive to obtain.
Multi-fidelity methods use a small set of high-fidelity data to enhance the
accuracy of a large set of low-fidelity data. In the approach described in this
paper, this is accomplished by constructing a graph Laplacian using the
low-fidelity data and computing its low-lying spectrum. This spectrum is then
used to cluster the data and identify points that are closest to the centroids
of the clusters. High-fidelity data is then acquired for these key points.
Thereafter, a transformation that maps every low-fidelity data point to its
bi-fidelity counterpart is determined by minimizing the discrepancy between the
bi- and high-fidelity data at the key points, and to preserve the underlying
structure of the low-fidelity data distribution. The latter objective is
achieved by relying, once again, on the spectral properties of the graph
Laplacian. This method is applied to a problem in solid mechanics and another
in aerodynamics. In both cases, this methods uses a small fraction of
high-fidelity data to significantly improve the accuracy of a large set of
low-fidelity data.
