Sensitivity analysis. Adjoint approach.
On the contrary to the direct approach, the adjoint approach starts from the Lagrangian formulation of the unconstrained optimization problem:
where F is the objective function, S is the set of state equations and u* are the discrete adjoint variables of the state variables acting as Lagrangian multipliers. Differentiating and solving for the adjoint variables, we obtain
Substituting in the derivative of the Lagrangian formulation gives:
The adjoint variables are evaluated once per state equation, making this the preferred approach for cases where the number of state equations is smaller than the number of design variables.
- Bletzinger, K.-U. (2017). Shape Optimization. In Encyclopedia of Computational Mechanics Second Edition (eds E. Stein, R. Borst and T.J.R. Hughes). https://doi.org/10.1002/9781119176817.ecm2109