On the solution of an inverse natural convection problem using various conjugate gradient methods

The inverse problem of determining the time-varying strength of a heat source, which causes natural con- vection in a two-dimensional cavity, is considered. The Boussinesq equation is used to model the natural convection induced by the heat source. The inverse natural convection problem is solved through the mini- mization of a performance function utilizing the conjugate gradient method. The gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct di erentiation method. The accuracy and e ciency of these two methods are compared, and a new method is suggested that exploits the advantageous aspects of both methods while avoiding the shortcomings of them