Minimax Error of Interpolation and Optimal Design of Experiments for Variable Fidelity Data

TL;DR

This paper derives minimax interpolation errors for variable fidelity Gaussian process regression, enabling optimal data sample allocation to maximize accuracy within computational budgets, and demonstrates improved performance over heuristics.

Contribution

It provides a theoretical framework for optimal design of experiments with variable fidelity data using minimax errors, guiding resource allocation for regression accuracy.

Findings

Optimal sample shares improve regression accuracy.

Variable fidelity data can outperform high fidelity only data.

Theoretical results are validated with real and synthetic data.

Abstract

Engineering problems often involve data sources of variable fidelity with different costs of obtaining an observation. In particular, one can use both a cheap low fidelity function (e.g. a computational experiment with a CFD code) and an expensive high fidelity function (e.g. a wind tunnel experiment) to generate a data sample in order to construct a regression model of a high fidelity function. The key question in this setting is how the sizes of the high and low fidelity data samples should be selected in order to stay within a given computational budget and maximize accuracy of the regression model prior to committing resources on data acquisition. In this paper we obtain minimax interpolation errors for single and variable fidelity scenarios for a multivariate Gaussian process regression. Evaluation of the minimax errors allows us to identify cases when the variable fidelity data provides better interpolation accuracy than the exclusively high fidelity data for the same computational budget. These results allow us to calculate the optimal shares of variable fidelity data samples under the given computational budget constraint. Real and synthetic data experiments suggest that using the obtained optimal shares often outperforms natural heuristics in terms of the regression accuracy.