  In an era where scientific experiments can be very costly, multi-fidelity
emulators provide a useful tool for cost-efficient predictive scientific
computing. For scientific applications, the experimenter is often limited by a
tight computational budget, and thus wishes to (i) maximize predictive power of
the multi-fidelity emulator via a careful design of experiments, and (ii)
ensure this model achieves a desired error tolerance with some notion of
confidence. Existing design methods, however, do not jointly tackle objectives
(i) and (ii). We propose a novel stacking design approach that addresses both
goals. A multi-level reproducing kernel Hilbert space (RKHS) interpolator is
first introduced to build the emulator, under which our stacking design
provides a sequential approach for designing multi-fidelity runs such that a
desired prediction error of $\epsilon > 0$ is met under regularity assumptions.
We then prove a novel cost complexity theorem that, under this multi-level
interpolator, establishes a bound on the computation cost (for training data
simulation) needed to achieve a prediction bound of $\epsilon$. This result
provides novel insights on conditions under which the proposed multi-fidelity
approach improves upon a conventional RKHS interpolator which relies on a
single fidelity level. Finally, we demonstrate the effectiveness of stacking
designs in a suite of simulation experiments and an application to finite
element analysis.
