Convex relaxation for IMSE optimal design in random-field models

The 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.