Stacking designs: designing multi-fidelity computer experiments with target predictive accuracy

TL;DR

This paper introduces a stacking design method for multi-fidelity experiments that optimizes predictive accuracy within a computational budget, providing theoretical guarantees and demonstrating practical effectiveness.

Contribution

It proposes a novel stacking design approach combined with a multi-level RKHS interpolator to jointly optimize accuracy and cost in multi-fidelity experiments.

Findings

The stacking design achieves desired prediction error with bounded cost.

Theoretical cost complexity bounds are established for the proposed method.

Empirical results show improved efficiency over traditional single-fidelity approaches.

Abstract

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.