Bi-fidelity sparse-grid interpolation driven by a local-error estimator

TL;DR

This paper introduces a bifidelity sparse-grid interpolation method that adaptively refines the approximation using a local-error estimator, reducing computational costs while maintaining accuracy for high-dimensional, complex functions.

Contribution

It presents a novel incremental sparse grid strategy that combines local error estimation with bifidelity predictions to improve efficiency in high-dimensional function approximation.

Findings

Effective in approximating high-dimensional functions with localized features.

Reduces computational costs by selectively evaluating the target function.

Demonstrated success on benchmark functions and an engineering application.

Abstract

Sparse grids based on Lagrange polynomials have become one of the staple methods for approximating functions that are high-dimensional and expensive to evaluate, in the context e.g. of PDE-based parametric design exploration. They are however known to be inefficient for problems requiring local refinement, such as when the target function exhibits localized features or sharp gradients. While locally-refined sparse grids based e.g. on piecewise linear polynomials are a well-established alternative to circumvent this problem, in this work we present a strategy for improving the local efficiency of Lagrangian sparse grids. We do so by building the sparse grid approximation incrementally and evaluating the function only at collocation points at which a suitable (and crucially, zero-cost) error indicator suggest that incorporating the function evaluation would significantly change the landscape of the approximation. The remaining collocation points are instead assigned values predicted by the already available sparse grid, i.e., following a bifidelity approach that reduces costs while preserving accuracy. The effectiveness of this methodology is demonstrated on several benchmark analytical functions and an engineering application concerning flashback phenomena in hydrogen-fueled perforated burners.