Optimal design for kernel interpolation: applications to uncertainty quantification

TL;DR

This paper introduces a quasi-optimal kernel interpolation method with improved stability for uncertainty quantification, outperforming sparse grid methods and applicable to gradient-enhanced Gaussian process emulators.

Contribution

It proposes a new procedure for selecting interpolation points that reduces condition numbers, enhancing stability and accuracy in kernel interpolation for uncertainty quantification.

Findings

Quasi-optimal points reduce interpolation instability.

Method outperforms sparse grid approaches in several cases.

Applicable to gradient-enhanced Gaussian process modeling.

Abstract

The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of \textit{candidate} points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points result in a smaller condition number, and thus mitigates the instability of the interpolation procedure when the number of points becomes large. Applications to parametric uncertainty quantification are presented, and it is shown that the proposed interpolation method can outperform sparse grid methods in many interesting cases. We also demonstrate the new procedure can be applied to constructing gradient-enhanced Gaussian process emulators.