A manifold of planar triangular meshes with complete Riemannian metric

Herzog, Roland; Loayza-Romero, Estefanía

doi:10.1090/mcom/3775

Cited by 4 publications

(1 citation statement)

References 35 publications

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“…In [25], a shape space is just modeled as a linear (vector) space, which in the simplest case is made up of vectors of landmark positions. However, there is a large number of different shape concepts, e.g., plane curves [37], surfaces in higher dimensions [3,36], boundary contours of objects [30,51], multiphase objects [50], characteristic functions of measurable sets [52], morphologies of images [13], and planar triangular meshes [18]. The choice of the shape space depends on the demands in a given situation.…”

Section: Newton Derivative Calculusmentioning

confidence: 99%

A Newton derivative scheme for shape optimization problems constrained by variational inequalities

Goldammer¹,

Schulz²,

Welker³

2022

Preprint

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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 fast and higher order solution algorithms. In this paper, we consider Newton-derivatives in order to formulate optimality conditions. In this context, we set up a Newton-shape derivative scheme. Examples show the application of the proposed scheme.

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Section: Newton Derivative Calculusmentioning

confidence: 99%

A Newton derivative scheme for shape optimization problems constrained by variational inequalities

Goldammer¹,

Schulz²,

Welker³

2022

Preprint

View full text Add to dashboard Cite

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A Discretize-then-Optimize Approach to PDE-Constrained Shape Optimization

Herzog,

Loayza-Romero

2024

ESAIM: COCV

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We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see [14]. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of globally optimal solutions for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.

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Riemannian geometry for efficient analysis of protein dynamics data

Diepeveen,

Esteve-Yagüe,

Lellmann

et al. 2024

Proc. Natl. Acad. Sci. U.S.A.

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An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

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A manifold of planar triangular meshes with complete Riemannian metric

Cited by 4 publications

References 35 publications

A Newton derivative scheme for shape optimization problems constrained by variational inequalities

A Newton derivative scheme for shape optimization problems constrained by variational inequalities

A Discretize-then-Optimize Approach to PDE-Constrained Shape Optimization

Riemannian geometry for efficient analysis of protein dynamics data

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