We address the problem of computing 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 be easily 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.

International audienceThe construction of optimal designs for random-field interpolation models via convex design theory is considered. The definition of an Integrated Mean-Squared Error (IMSE) criterion yields a particular Karhunen–Loève expansion of the underlying random field. After spectral truncation, the model can be interpreted as a Bayesian (or regularised) linear model based on eigenfunctions of this Karhunen–Loève expansion, and can be further approximated by a linear model involving orthogonal observation errors. Using the continuous relaxation of approximate design theory, the search of an IMSE optimal design can then be turned into a Bayesian A-optimal design problem, which can be efficiently solved by convex optimisation. A careful analysis of this approach is presented, also including the situation where the model contains a linear parametric trend, which requires specific treatments. Several approaches are proposed, one of them enforcing orthogonality between the trend functions and the complementary random field. Convex optimisation, based on a quadrature approximation of the IMSE criterion and a discretisation of the design space, yields an optimal design in the form of a probability measure with finite support. A greedy extraction procedure of the exchange type is proposed for the selection of observation locations within this support, the size of the extracted design being controlled by the level of spectral truncation. The performance of the approach is investigated on a series of examples indicating that designs with high IMSE efficiency are easily obtained

We address the problem of computing IMSE (Integrated Mean-Squared Error) optimal designs for random fields interpolation with known mean and covariance. We both consider the IMSE and truncated-IMSE (approximation of the IMSE by spectral truncation). We assume that the MSE is integrated through a discrete measure and restrict the design space to the support of the considered measure. The IMSE and truncated-IMSE of such designs can be easily evaluated at the cost of some simple preliminary computations, making global optimization affordable. Numerical experiments are carried out and illustrate the interest of the considered approach for the approximation of IMSE optimal designs.