Heuristic linear algebraic rank-variance formulation and solution approach for efficient sensor placement

The digital age has significantly impacted our ability to sense our environment and infer the state or status of equipment in our environment from the sensed information. Consequently inferring from a set of observations the causal factors that produced them is known as an inverse problem. In this study the sensed information, a.k.a. sensor measurement variables, is measurable while the inferred information, a.k.a. target variables, is not measurable. The ability to solve an inverse problem depends on the quality of the optimisation approach and the relevance of information used to solve the inverse problem. In this study, we aim to improve the information available to solve an inverse problem by considering the optimal selection of m sensors from k options. This study introduces a heuristic approach to solve the sensor placement optimisation problem which is not to be confused with the required optimisation strategy to solve the inverse problem. The proposed heuristic optimisation approach relies on the rank of the cross-covariance matrix between the observations of the target variables and the observations of the sensor measurement variables obtained from simulations using the computational model of an experiment. In addition, the variance between observations of the sensor measurements is considered. A new formulation, namely the tolerance rank-variance formulation (TRVF) is introduced and investigated numerically on a full field deterioration problem. The full field deterioration is estimated for a plate by resolving a parametrisation of the deterioration field for four scenarios. We demonstrate that the optimal sensor locations not only depend on the loading and boundary conditions of the plate but also on the expected ranges for the deterioration parameters. Although the sensor placements are not provably optimal the numerical results clearly indicate computationally efficient near optimal sensor placements.