Editing Shape optimization (section)

=== Iterative methods using shape gradients ===

Consider a smooth velocity field <math>V</math> and the family of transformations <math>T_s</math> of the initial domain <math>\Omega_0</math> under the velocity field <math>V</math>:

:<math>x(0) = x_0 \in \Omega_0, \quad x'(s) = V(x(s)), \quad T_s(x_0) = x(s), \quad s \geq 0 </math>,

and denote

:<math>\Omega_0 \mapsto T_s(\Omega_0) = \Omega_s.</math>

Then the Gâteaux or shape derivative of <math>\mathcal{F}(\Omega)</math> at <math>\Omega_0</math> with respect to the shape is the limit of

:<math>d\mathcal{F}(\Omega_0;V) = \lim_{s \to 0}\frac{\mathcal{F}(\Omega_s) - \mathcal{F}(\Omega_0)}{s}</math>

if this limit exists. If in addition the derivative is linear with respect to <math>V</math>, there is a unique element of <math>\nabla \mathcal{F} \in L^2(\partial \Omega_0)</math> and

:<math>d\mathcal{F}(\Omega_0;V) = \langle \nabla \mathcal{F}, V \rangle_{\partial \Omega_0}</math>

where <math>\nabla \mathcal{F}</math> is called the shape gradient. This gives a natural idea of [[gradient descent]], where the boundary <math>\partial \Omega</math> is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional. Higher order derivatives can be similarly defined, leading to Newtonlike methods.

Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the [[Hessian matrix|Hessian]])  of the objective functional <math>\mathcal{F}</math>.

If the shape optimization problem has constraints, that is, the functional 
<math>\mathcal{G}</math> is present, one has to find ways to convert the 
constrained problem into an unconstrained one. Sometimes ideas based on [[Lagrange multipliers]], like the [[adjoint state method]], can work.