## Homotopy methods: connecting problems.

In this talk, we derive the basic principle of homotopy methods (also called continuation methods) and apply them to shape optimization problem.

Here, first order methods can be used. However, it is common that this methods require a large amount of iterations.

In this case, higher order methods can significantly reduce the number of iterations. But for many of those methods convergence can be just shown for a 'close enough' initial guess.

We continously connect the given problem to a so called simple problem. It is of advantage if the solution of the simple problem is either known or easily computable. Once the connection is established, we use path following techniques to follow to the connection. The path following technique presented in this work follows predictor corrector scheme.

Finally, the derived algorithms are applied to some numerical examples.