Convex relaxation for IMSE optimal design in random-field models

The de nition of an Integrated Mean-Squared Error (IMSE) criterion for the learning of a random eld model yields a particular Karhunen-Loève expansion of the underlying eld. The model can thus also be interpreted as a Bayesian (or regularised) linear model based on eigenfunctions of this Karhunen-Loève expansion, and can be 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 e ciently solved by convex optimisation. We propose a greedy extraction procedure, of the exchange type, that permits to select observation locations among support points of an optimal design measure. In the presence of a parametric trend, we show how speci c treatments can be applied to avoid confusion between the trend and eigenfunctions. The performance of the approach is investigated on a series of examples indicating that designs with very high IMSE-e ciency are easily obtained.