  Inspired by multi-fidelity methods in computer simulations, this article
introduces procedures to design surrogates for the input/output relationship of
a high-fidelity code. These surrogates should be learned from runs of both the
high-fidelity and low-fidelity codes and be accompanied by error guarantees
that are deterministic rather than stochastic. For this purpose, the article
advocates a framework tied to a theory focusing on worst-case guarantees,
namely Optimal Recovery. The multi-fidelity considerations triggered new
theoretical results in three scenarios: the globally optimal estimation of
linear functionals, the globally optimal approximation of arbitrary quantities
of interest in Hilbert spaces, and their locally optimal approximation, still
within Hilbert spaces. The latter scenario boils down to the determination of
the Chebyshev center for the intersection of two hyperellipsoids. It is worth
noting that the mathematical framework presented here, together with its
possible extension, seems to be relevant in several other contexts briefly
discussed.
