  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.
