In certain data assimilation and optimization problems, gradient information is essentially required. For this purpose, the adjoint model (ADJM) is often employed. The ADJM is a transpose of the tangent linear model (TLM); thus, both are based on the tangent linear approximation of the corresponding nonlinear model. Derivations, formulations, and correctness checks of the TLM and ADJM are described in detail along with the construction of the practical codes of the TLM/ADJM. Practical methods of deriving the ADJM are introduced, using the adjoint operator, Lagrangian multipliers, and chain rules. Uncertainty and validity of the TLM/ADJM are also discussed in terms of nonlinearity and discontinuous physical processes in numerical models. An example of deriving the ADJM is given for Burgers’ equation by comparing the adjoint operator method and the Lagrangian multipliers methods.

At the opposite extreme to the results of the last chapter, we now consider unbounded operators on Hilbert spaces. We define an appropriate notion of the Hilbert adjoint in this setting and show that for unbounded self-adjoint operators, we can develop a good spectral theory.