Spectral Approximation of the IMSE Criterion for Optimal Designs in Kernel-Based Interpolation Models

We address the problem of computing integrated mean-squared error (IMSE)-optimal designs for random field interpolation models. A spectral representation of the IMSE criterion is obtained from the eigendecomposition of the integral operator defined by the covariance kernel of the random field and integration measure considered. The IMSE can then be approximated by spectral truncation, and bounds on the error induced by this truncation are given. We show how the IMSE and truncated-IMSE can easily be computed when a quadrature rule is used to approximate the integrated MSE and the design space is restricted to a subset of quadrature points. Numerical experiments are carried out and indicate (i) that retaining a small number of eigenpairs (in regard to the quadrature size) is often sufficient to obtain good approximations of IMSE-optimal quadrature-designs when optimizing the truncated criterion and (ii) that optimal quadrature-designs generally give efficient approximations of the true optimal designs for the quadrature approximation of the IMSE.