Computing Gradients by Adjoint States

Abstract

We have already observed that the key point in sensitivity analysis or in most optimization algorithms is the accurate calculation of the gradient of the cost functionj.In order to detail this calculation, it is useful to first recall the structure of a cost function in shape optimization. In such problems, the cost function is defined by

$$\left\{ {\begin{array}{*{20}{c}} {j:{Z_{zdm}} \subset {\mathbb{R}^n} \to \mathbb{R},} \\ {z \mapsto j\left( z \right) = J\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} \left( z \right),W\left( z \right)} \right),} \end{array}} \right.$$

the state variable W = W(z) being the solution of a (direct) system of state equations

$${f_h}\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} \left( z \right),W} \right) = 0in{\mathbb{R}^\mathcal{P}}$$

(6.1)

which are themselves functions of the present positionX(z)of the computational grid.