Statistical identification of sensors location using inverse differential equations

Many engineering designs are built on the basis of differential equations. In some cases, parameters of the underlying differential equations are not known and should be estimated from the measurement data at the sensors´ locations, leading to an inverse problem. In this paper, we demonstrate how the optimal locations of these sensors can be determined using the statistical theory of optimal experiments. Three specific engineering problems are discussed to illustrate the statistical criteria: (1) adaptive determination of optimal locations of temperature sensors in a one-dimensional inverse heat transfer problem, (2) localization of a power leak/damage in conductive wire, (3) adaptive design in the framework of a one-dimensional boundary-value problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal designs are applied.