E-optimum sensor selection for estimation of subsets of parameters

Abstract

The design of a network of observation locations is addressed in the setting of estimating unknown parameters of a spatiotemporal system modelled by a partial differential equation. Interest is in estimating only a subset of these parameters as accurately as possible. The other parameters, called nuisance parameters, must also be estimated although we are interested in neither their values, nor accuracies. The maximal eigenvalue of the covariance matrix of the maximum-likelihood estimator of the parameters of interest is used as the measure of the identification accuracy. In order to make selection of a best subset of gauged sites from a possibly very large set of candidate sites computationally tractable, its convex relaxation is introduced. Two major problems to be tackled are the potential singularity of the optimal information matrix associated with all unknown parameters and the nondifferentiability of the optimality criterion. The former is settled by imposing a constraint on the minimal allowable value of the determinant of the information matrix. The latter is resolved by reformulating the problem as a convex semi-infinite programming problem whose solution is sought by solving a sequence of finite low-dimensional min–max problems using extremely efficient generalized simplicial decomposition. The excellent performance of the proposed technique is illustrated by an example involving optimal sensor node activation in a large sensor network collecting measurements to identify a moving contaminating source.