The distributed expression of the shape derivative, also known as volumetric form or weak form of the shape derivative, allows to study shape calculus and optimization of nonsmooth geometries such as Lipschitz domains and polygons. The distributed shape derivative has witnessed a surge of interest recently, in particular due to new ideas for numerical applications. We will discuss some recent theoretical developments for distributed shape derivatives, and their application to level set methods in numerical shape optimization. We will then show some applications to inverse problems.
This talk is devoted to the inverse problem of recovering the unknown distributed flux on an inaccessible part of boundary using measurement data on the accessible part. We establish and verify a variational source condition for this inverse problem, leading to a logarithmic-type convergence rate for the corresponding Tikhonov regularization method under a low Sobolev regularity assumption on the distributed flux. Our proof is based on the conditional stability and Carleman estimates together with the complex interpolation theory on a proper Gelfand triplet
We discuss a shape optimization problem subject to an elliptic curl-curl VI of the second kind. By employing a Moreau-Yosida-type regularization to its dual formulation, we obtain the sufficient differentiablity property to apply the averaged adjoint method. By means of this, we compute the shape derivative of the regularized problem. After presenting a stability estimate for the shape derivative w.r.t. the regularization parameter, we prove the strong convergence of the optimal shapes and the solutions. Finally, we apply the numerical algorithm to a problem from the (HTS) superconductivity.