The aim of this paper is to open up a new perspective on set and shape optimization by establishing a theory of Galerkin approximations to the space of convex and compact subsets of \({\mathbb {R}}^d\) with favorable properties, both from a theoretical and from a computational perspective. Galerkin spaces consisting of polytopes with fixed facet normals are first explored in depth and then used to solve optimization problems in the space of convex and compact subsets of \({\mathbb {R}}^d\) approximately.

中文翻译：

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在$$ {\ mathbb {R}} ^ d $$ R d的凸集和紧集子空间中进行优化的Galerkin方法

本文的目的是通过建立具有良好特性的\（{\ mathbb {R}} ^ d \）的凸集和紧集的空间的Galerkin逼近理论，为设置和形状优化开辟新的视角，从理论和计算的角度来看。首先深入研究由固定多面法线的多面体组成的Galerkin空间，然后将其用于解决\（{{mathbb {R}} ^ d \）的凸集和紧集的空间中的优化问题。