A tutorial on the adjoint method for inverse problems

This paper is a basic tutorial on the adjoint method when used in a computational scheme for solving an inverse problem. The adjoint method is a technique for the efficient calculation of the gradient of the functional which is to be minimized in the solution process. The method is presented in a slightly non-standard way, which is believed to be simpler and less abstract than the common presentation found in most books and papers, yet equally general. More specifically, the unknown parameters are discretized from the outset while all other variables remain continuous. The adjoint method is applied here both at the continuous level and at the discrete level. Both steady-state (elliptic) and time-dependent problems are considered. Various computational aspects are discussed.