Adjoint state method

Summary

The adjoint state method computes the gradient of an objective that depends on parameters through the solution of an equation. A typical example of such an equation is a discretised differential equation. Instead of differentiating the state with respect to each parameter, it solves one extra linear equation, the adjoint equation, whose solution is the adjoint state. The cost of the gradient is then about that of one or two state solves, whatever the number of parameters. It is the basis of PDE-constrained optimization, of seismic and medical inversion, and of reverse-mode differentiation.