Shape optimization is concerned with identifying shapes, or subsets of $\mathbb{R}$, behaving in an optimal way with respect to a given physical system. Many problems of interest involve a system in the form of a partial differential equation (PDE), the solution of which depends on one or more shapes defining the domain. Theory and algorithms in shape optimization can be based on techniques from differential geometry, e.g., a Riemannian manifold structure can be used to define the distances of two shapes. Thus, shape spaces are of particular interest in shape optimization.
In this talk, we apply the differential-geometric structure of Riemannian shape spaces to the theory of deterministic shape optimization problems. We also present a space containing shapes in $\mathbb{R}^2$ that can be identified with a Riemannian product manifold but at the same time admits piecewise smooth curves as elements. Since many relevant problems in the area of shape optimization involve a constraint in the form of a PDE, which contains inputs or material properties that may be unknown or subject to uncertainty, we consider also stochastic shape optimization problems. We present algorithms to solve deterministic and stochastic (multi-)shape optimization problems and give numerical results of these algorithms.