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

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.