Optimal design for kernel interpolation: Applications to uncertainty quantification

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 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 results in 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.

POSYDON: A General-purpose Population Synthesis Code with Detailed Binary-evolution Simulations

“…Instead, these two could be combined into a joint treatment to reduce errors, making additional use of covariances between the different grid types (Singh et al 2016). We will also explore adopting kernelizedinterpolation approaches (i.e., Wilson & Nickisch 2015;Gardner et al 2018;Narayan et al 2021). To illustrate the use of such an approach, it is observed in Figure 22 that the confusion matrix for our grid of He-rich stars with CO companions has a lower accuracy for unstable mass transfer.…”

POSYDON: A General-Purpose Population Synthesis Code with Detailed Binary-Evolution Simulations

“…Instead these two could be combined into a joint treatment to reduce errors, making additional use of covariances between the different grid types (Singh et al 2016). We will also explore adopting kernelized-interpolation approaches (i.e., Wilson & Nickisch 2015;Gardner et al 2018;Narayan et al 2021). To illustrate the use of such an approach, it is observed in Figure 22 that the confusion matrix for our grid of He-rich stars with compact object companions has a lower accuracy for unstable mass transfer.…”

Stable Likelihood Computation for Machine Learning of Linear Differential Operators With Gaussian Processes

“…One of the major fields in applied sciences is to model different phenomena in terms of flexible operational equations [1,2]. In other words, the researchers usually attempt to find a coherent form of flexible operational equations corresponding to the observed data to the effect that they best describe and govern them [3,4]. Therefore, the necessity of existence of parametric operational equations is created, that is, the equations that have parameters and they increase the flexibility to cover the observed data.…”