Seminar in Numerical Analysis: Stephan Schmidt (Universität Würzburg)

SQP Methods for Shape Optimization Based on Weak Shape Hessians

Many PDE constrained optimization problems fall into the category of shape optimization, meaning the geometry of the domain is the unknown to be found. Most natural applications are drag minimization in fluid dynamics, but many tomography and image reconstruction problems also fall into this category.

The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced.

The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

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