Recently, the level set method has received growing attention as a
tool to solve inverse obstacle and shape optimization problems.
So far, the majority of algorithms in this subject were based on
gradient-type methods We discuss the coupling of level set methods with fast optimization techniques such as Newton's method or sequentially quadratic programming methods.
Through several applications to inverse obstacle problems we highlight the main theoretical and numerical issues. Finally, we
demonstrate the improvement with respect to gradient-based methods in numerical experiments.
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