Optimal design for linear models via gradient flow

Abstract

Optimal experimental design (OED) aims to choose the observations in an experiment to be as informative as possible, according to certain statistical criteria. In the linear case (when the observations depend linearly on the unknown parameters), it seeks the optimal weights over rows of the design matrix $\mA$ under certain criteria. Classical OED assumes a discrete design space and thus a design matrix with finite dimensions. In many practical situations, however, the design space is continuous-valued, so that the OED problem is one of optimizing over a continuous-valued design space. The objective becomes a functional over the probability measure, instead of a function of a finite dimensional vector. This change of perspective requires a new set of techniques to optimize over probability measures, and Wasserstein gradient flow becomes a natural candidate. Both the first-order criticality and the convexity properties of the OED objective are presented. Computationally, the Monte Carlo particle method is used to translate the gradient flow equation formulation into a numerical algorithm. This algorithm is applied to two elliptic inverse problems.