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Optimal design for kernel interpolation: applications to uncertainty quantification. (English) Zbl 07506525

Summary: 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.

MSC:

41Axx Approximations and expansions
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Nxx Numerical methods for partial differential equations, boundary value problems

Software:

GaussQR; Matlab
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Full Text: DOI arXiv

References:

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