Adjoint state method

The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It is applied in geophysics, seismic imaging and photonics, and more recently in neural networks. The method treats the equation that defines the state of a system as a constraint on the quantity being minimized, forms the Lagrangian, and introduces an adjoint state variable that is the Lagrange multiplier. Solving the adjoint state equation, which involves the adjoint of the operator in the state equation, gives the gradient with respect to the parameters from a direct solve and one adjoint solve. A useful property is that the amount of computation does not depend on the number of parameters whose gradient is wanted. The original adjoint calculation goes back to Jean Cea, who used the Lagrangian to compute derivatives of a functional with respect to a shape parameter, and the approach is also used in the Landweber iteration method.