Convergence of a Newton algorithm for semi-discrete optimal transport

Kitagawa, Jun; Mérigot, Quentin; Thibert, Boris

doi:10.4171/jems/889

Cited by 96 publications

(211 citation statements)

References 24 publications

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“…. As proved in [17], the constant C appearing in (Diff-2-b) depends on the minimal angle between the intersection of two competition zones e ik and e il . This constant is non-zero since there is a finite number of such intersections and it drives the C 2,α regularity of the function g.…”

Section: The Euclidean Casementioning

confidence: 88%

“…Letting s goes to zero shows that lim 0 + r(t) exists and is equal to α f (0) which proves lemma 1.1. Proof of lemma 1.2 The proof of this proposition owes so much to [17], proposition 3.2 that we even take the same notations. Consider the following partition of Li(t) ∩ L k (t) :…”

Section: Proof Of Lemmatamentioning

confidence: 96%

“…The differentiation with respect to m is rather straightforward and will not be discussed hereafter. The second order differentiability of g with respect to φ is known [21,7,18] and proved in [17]. This proof mainly uses that the Laguerre cells Li are the intersection for all j = i of the sub-level sets (with respect to the value of…”

Section: Semi-discrete Optimal Transportmentioning

confidence: 98%

“…However he has given no proof of existence. We will make use of the framework developed by Kitagawa and Mérigot [17] to overcome these restrictions.…”

Section: Introductionmentioning

confidence: 99%

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Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

2018

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Section: The Euclidean Casementioning

confidence: 88%

Section: Proof Of Lemmatamentioning

confidence: 96%

Section: Semi-discrete Optimal Transportmentioning

confidence: 98%

“…However he has given no proof of existence. We will make use of the framework developed by Kitagawa and Mérigot [17] to overcome these restrictions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

2018

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“…Lebesgue while the other one is discrete, the transport problem can be reduced to a finite-dimensional convex problem whose number of unknowns scales with the cardinality of the support of the discrete distribution. Leveraging tools from computational geometry, this semi-discrete problem can be solved efficiently up to fairly large scale when the cost is Euclidean [Aurenhammer et al 1998;De Goes et al 2012;Kitagawa et al 2016;Lévy 2015;Mérigot 2011].…”

Section: Semi-discrete Optimal Transportmentioning

confidence: 99%

Dynamical optimal transport on discrete surfaces

2018

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Fig. 1. Given two probability distributions over a discrete surface (left and right), our algorithm generates an interpolation that takes the geometric structure of the surface into account.We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.

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Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

Schneider

Wicht

2022

Numerical Meth Engineering

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We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial-scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning.

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Convergence of a Newton algorithm for semi-discrete optimal transport

Cited by 96 publications

References 24 publications

Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

Dynamical optimal transport on discrete surfaces

Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

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