Towards a Hybrid Adjoint Approach for Arbitrarily Complex Partial Differential Equations

Adjoint methods are widely used in various areas of computational science to efficiently obtain sensitivities of functionals which result from the solution of partial differential equations (PDEs). In addition, adjoint methods have been used in other settings including error estimation, uncertainty quantification and inverse problem formulations. When deriving the adjoint equations, there are two main approaches one can follow: the discrete and the continuous methods, which differ principally in the order of the linearization and discretization steps. The discrete adjoint method starts from the discretized form of the partial differential equation, which is then linearized. On the other hand, the continuous method linearizes the continuous governing equations first and then discretizes the resulting problem. Each of these approaches is found to have advantages and disadvantages over the other. In this paper we consider a hybrid approach between these two methods that aims to combine the better qualities of both: reducing the time spent on the mathematical derivation while also lowering the computational requirements of the discrete method and increasing the overall quality of the adjoint solution.