Towards optimization techniques on diffeological spaces

Goldammer, Nico; Welker, Kathrin

doi:10.1002/pamm.202000040

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…In contrast to shape spaces as Riemannian manifolds, research for diffeological spaces as shape spaces has just begun, see e.g. [34,99]. Therefore, for the following section we will focus on Riemannian manifolds first, and then briefly consider diffeological spaces.…”

Section: Different Points In the Space Have Different Neighborhoods (...mentioning

confidence: 99%

“…Metrics for diffeological spaces have been researched to a lesser extent. However most concepts can be transferred, and in [34] a Riemannian metric is defined for a diffeological space, which yields a Riemannian diffeological space. Additionally, the Riemannian gradient and a steepest descent method on diffeological spaces are defined, assuming a Riemannian metric is available.…”

Section: Metrics On Shape Spacesmentioning

confidence: 99%

“…the generalization of concepts like the definition of a gradient, a distance measure and optimality conditions, is not yet reached for the novel space B 1 2 (Γ 0 ). However, the necessary objects for the steepest descent method on a diffeological space are established and the corresponding algorithm is formulated in [34]. It is nevertheless worth to mention that various numerical experiments, e.g.…”

Section: Examples Of Shape Spaces and Their Use For Shape Optimizationmentioning

confidence: 99%

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Parameter-free shape optimization: various shape updates for engineering applications

Radtke¹,

Bletsos²,

Kühl³

et al. 2023

Preprint

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In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims at an explanation of several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered.

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Section: Different Points In the Space Have Different Neighborhoods (...mentioning

confidence: 99%

Section: Metrics On Shape Spacesmentioning

confidence: 99%

Section: Examples Of Shape Spaces and Their Use For Shape Optimizationmentioning

confidence: 99%

See 1 more Smart Citation

Parameter-free shape optimization: various shape updates for engineering applications

Radtke¹,

Bletsos²,

Kühl³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Question: Which tangent space definition is appropriate for optimization techniques on diffeological spaces? Answer: None of the above-mentioned definitions is suitable for formulating optimization methods ⇒ Definition of a novel one in [2].…”

Section: Challenge: Tangent Spacementioning

confidence: 99%

Optimization on diffeological spaces

Goldammer

Welker

2021

Proc Appl Math and Mech

Self Cite

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On this poster, we present optimization techniques on diffeological spaces. Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. In order to generalize optimization methods known on manifolds to diffeological spaces, we define various objects like a diffeological tangent space, a diffeological Riemannian space as well as a diffeological gradient. In addition we give the definition of a diffeological retraction. These objects are necessary for formulating the steepest descent method on diffeological spaces. We present this method and apply it on an example.

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Elastic diffeological spaces

Blohmann

2024

Contemporary Mathematics

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We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosický. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological groups and diffeological vector spaces with a mild extra condition, mapping spaces between smooth manifolds, and spaces of sections of smooth fiber bundles.

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Towards optimization techniques on diffeological spaces

Cited by 4 publications

References 6 publications

Parameter-free shape optimization: various shape updates for engineering applications

Parameter-free shape optimization: various shape updates for engineering applications

Optimization on diffeological spaces

Elastic diffeological spaces

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