Optimization of convex shapes: An approach to crystal shape identification

Abstract

We consider a shape identification problem of growing crystals. The shape of the crystal is to be constructed from a single interferometer measurement. This is an ill-posed inverse problem. The forward problem of interferogram from shape is injective if we restrict the problem to convex shapes with known boundary. The problem is formulated as a shape optimization problem. Our aim is to solve this numerically using the gradient descent method. In the numerical computations of this paper we study the behavior of the approach in simplified cases. Using H 1-gradients (inner products) acts as a regularization method. Methods for enforcing the convexity of shapes are discussed. © 2009 Springer Berlin Heidelberg.