Sensor Selection for Minimizing the Condition Number with Guaranteed E-Efficiency

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. The resulting measurements are supposed to make the Hessian of the least-squares criterion well-conditioned, which amounts to the design yielding a minimal condition number of the corresponding Fisher information matrix (FIM). To prevent this matrix from being close to singular, an additional constraint is imposed on the lowest allowable value of its largest eigenvalue. In order to make selection of a best subset of gauged sites from a possibly very large set of candidate sites computationally tractable, its relaxed version is introduced in the form of a fractional programming problem. Using the Charnes-Cooper variable transformation, it is converted to an equivalent convex problem. Its nonsmoothness is then circumvented by expressing the problem as a semi-infinite programming problem. It is then solved by an exchange method, which reduces to an extremely simple algorithm that alternates between finding the eigenvalues and eigenvectors of the current FIM and solving a linear-programming problem. The excellent performance of the proposed technique is illustrated by a nontrivial simulation example.