On computation of the design function gradient for the sensor-location problem in variational data assimilation

The optimal sensor-location problem is considered in the framework of variational data assimilation for a large-scale dynamical model governed by partial differential equations. This problem is formulated as an optimization problem for the design function defined on the limited memory approximation of the inverse Hessian of the data assimilation cost function. The expression for the gradient of the design function with respect to the sensor-location coordinates is derived via the adjoint to the Hessian derivative. An efficient algorithm for the gradient evaluation suitable for large-scale applications is suggested. This algorithm exploits the special structure of the limited memory inverse Hessian defined by a small number of Ritz pairs obtained by the Lanczos method. If additional memory is allocated and certain data are stored during the computation of the Ritz pairs, no additional runs of the tangent linear model are required to evaluate the gradient. The accuracy of the gradients is checked in the numerical experiments. These gradients can be used for the gradient-based optimization of the design function within the chosen global optimization procedure.