Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo, Giuseppe; Pascale, Luigi De

doi:10.1007/978-3-540-44857-0_2

Cited by 6 publications

(4 citation statements)

References 104 publications

(65 reference statements)

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“…Since their interest for the Monge-Kantorovich problem and related problems, this kind of question was already studied in previous papers. So, a part of our results are well known by now and may exist in the literature in a more general setting and proved by using sophisticated arguments (see for instance [6,7] and the references therein). Our aim here is to give simple and direct proofs for our (more or less) simple situation.…”

Section: Introduction and Main Resultsmentioning

confidence: 59%

“…The connection between (7) and optimal transportation has been proved in [1] by using the equivalent formulation (2), where the tangential gradient introduced in [12] (see also [5,8] and the references therein) plays a fundamental role. For the regularity of m with respect to additional assumptions on µ, we refer the reader to the papers [13,14], [15,16] and the references therein.…”

Section: -546xmentioning

confidence: 99%

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Equivalent formulations for Monge–Kantorovich equation

Igbida

2009

Nonlinear Analysis: Theory, Methods & Applications

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Section: Introduction and Main Resultsmentioning

confidence: 59%

Section: -546xmentioning

confidence: 99%

Equivalent formulations for Monge–Kantorovich equation

Igbida

2009

Nonlinear Analysis: Theory, Methods & Applications

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“…In this section, we formulate the stochastic shape optimization model, which we use to demonstrate algorithm 1. The problem under consideration is an interface identification problem and has been studied in a number of texts [8,26,52]. A motivation for this model is in electrical impedance tomography, where the material distribution of electrical properties such as electric conductivity and permittivity inside the body is to be determined [9,29].…”

Section: Application To An Interface Identification Problemmentioning

confidence: 99%

Stochastic approximation for optimization in shape spaces

Geiersbach¹,

Loayza-Romero²,

Welker³

2020

Preprint

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In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the method on Riemannian manifolds and then make the connection to shape spaces. The method is demonstrated on a model shape optimization problem from interface identification. Uncertainty arises in the form of a random partial differential equation, where underlying probability distributions of the random coefficients and inputs are assumed to be known. We verify some conditions for convergence for the model problem and demonstrate the method numerically.

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“…We consider a model interface identification problem, which has been studied in the deterministic setting in a number of texts [5,12,26], and which we modify For the model, we concentrate on one-dimensional smooth shapes. In [16], the set of all one-dimensional smooth shapes is characterized by…”

Section: Model Formulationmentioning

confidence: 99%

Computational Aspects for Interface Identification Problems with Stochastic Modelling

Geiersbach,

Loayza,

Welker

2019

Preprint

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In this paper, a shape optimization problem constrained by a random elliptic partial differential equation with a pure Neumann boundary is presented. The model is motivated by applications in interface identification, where we assume coefficients and inputs are subject to uncertainty. The problem is posed as a minimization of the expectation of a random objective functional depending on the uncertain parameters. A numerical method for iteratively solving the problem is presented, which is a generalization of the classical stochastic gradient method in shape spaces. Moreover, we perform numerical experiments, which demonstrate the effectiveness of the algorithm.

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Optimal Shapes and Masses, and Optimal Transportation Problems

Cited by 6 publications

References 104 publications

Equivalent formulations for Monge–Kantorovich equation

Equivalent formulations for Monge–Kantorovich equation

Stochastic approximation for optimization in shape spaces

Computational Aspects for Interface Identification Problems with Stochastic Modelling

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