arXiv subjects

Nico Goldammer

Publications and source records attributed to Nico Goldammer.

3 recordsLinked to original sources

arXiv2024

Gâteaux semiderivative approach applied to shape optimization for contact problems

Shape optimization problems constrained by variational inequalities (VI) are non-smooth and non-convex optimization problems. The non-smoothness arises due to the variational inequality constraint, which makes it challenging to derive optimality conditions. Besides the non-smoothness there are complementary aspects due to the VIs as well as distributed, non-linear, non-convex and infinite-dimensional aspects due to the shapes which complicate to set up an optimality system and, thus, to develop efficient solution algorithms. In this paper, we consider Gâteaux semiderivatives in order to formulate optimality conditions. In the application, we concentrate on a shape optimization problem constrained by the contact problem.

math.OC↗

arXiv2023

On diffeologies from infinite dimensional geometry to PDE constrained optimization

We review how diffeologies complete the settings classically used from infinite dimensional geometry to partial differential equations, based on classical settings of functional analysis and with classical mapping spaces as key examples. As the classical examples of function spaces, we deal with manifolds of mappings in Sobolev classes (and describe the ILB setting), jet spaces and spaces of triangulations, that are key frameworks for the two fields of applications of diffeologies that we choose to highlight: evolution equations and integrable systems, and optimization problems constrained by partial differential equations.

math.DG↗

arXiv2021

Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts

Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. However, optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. This paper is devoted to an optimization technique on diffeological spaces. Thus, one main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need to formulate an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. We give examples for the novel objects and apply the presented diffeological algorithm to an optimization problem.

math.OC↗