Application of scaled nonlinear conjugate-gradient algorithms to the inverse natural convection problem

The inverse natural convection problem (INCP) in a porous medium is a highly non-linear problem because of the nonlinear convection and Forchheimer terms. The INCP can be converted into the minimization of a least-squares discrepancy between the observed and the modelled data. It has been solved using different classical optimization strategies that require a monotone descent of the objective function for recovering the unknown profile of the timevarying heat source function. In this investigation we use this PDE-constrained optimization problem as a demanding testbed to compare the performance of several state-of-the-art variants of the conjugate gradients approach. We propose solving the INCP using the scaled nonlinear conjugate gradient (SCG) method: a low-cost and low-storage optimization technique. The method presented here uses the gradient direction with a particular spectral step length and the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) updating formula without any matrix evaluations. Two adaptive line search approaches are numerically studied in which there is no need for solving the sensitivity problem to obtain the step length directly, and are compared to an exact line search approach. We combine the proposed optimization scheme for INCP with a consistent splitting scheme for solving systems of momentum, continuity and energy equations and a mixed finite element method. We show a number of computational tests which demonstrate that the proposed method performs better than the classical gradient method by improving the number of iterations required and reducing the computational time. We also discuss some practical issues related to the implementation of the different methods.