The second order perturbation approach for elliptic partial differential equations on random domains

Harbrecht, Helmut; Peters, Michael

doi:10.1016/j.apnum.2017.11.002

Cited by 10 publications

(9 citation statements)

References 22 publications

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“…Of course, according to (4), we only need the second order local shape derivative in case of identical fields, i.e., V = V . Hence, in order to compute the boundary conditions of the aforementioned equation (6)…”

Section: Shape Calculus For Parametrized Domainsmentioning

confidence: 99%

“…In order to compute the quantities E[u ε ], Cor[u ε ], and Cov[u ε ], appearing in (9), (10), and (11), we have to solve for E[δ 2 u] and Cor [δu]. To this end, we combine (6) and (8) to arrive at, see also [7],…”

Section: Statistical Moments On Random Domainsmentioning

confidence: 99%

“…Proof. The first two expressions have already been provided in [6,Lemma 2.3], whereas the third one is analogously derived by exploiting the trilinearity of δ 3 u 0 .…”

Section: Statistical Moments On Random Domainsmentioning

confidence: 99%

“…The perturbation approach, see e.g. [4,5,6,7], is motivated by small disturbances in manufacturing processes and models uncertain small deformations under the following view point. Notice that both approaches have recently been combined in [8].…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Dölz

Harbrecht

2018

Journal of Computational Physics

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Computing statistical quantities of interest of the solution of PDE on random domains is an important and challenging task in engineering. We consider the computation of these quantities by the perturbation approach. Especially, we discuss how third order accurate expansions of the mean and the correlation can numerically be computed. These expansions become even fourth order accurate for certain types of boundary variations. The correction terms are given by the solution of correlation equations in the tensor product domain, which can efficiently be computed by means of H-matrices. They have recently been shown to be an efficient tool to solve correlation equations with rough data correlations, that is, with low Sobolev smoothness or small correlation length, in almost linear time. Numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm. The application to a non-smooth domain is also included.

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Section: Shape Calculus For Parametrized Domainsmentioning

confidence: 99%

Section: Statistical Moments On Random Domainsmentioning

confidence: 99%

“…Proof. The first two expressions have already been provided in [6,Lemma 2.3], whereas the third one is analogously derived by exploiting the trilinearity of δ 3 u 0 .…”

Section: Statistical Moments On Random Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Dölz

Harbrecht

2018

Journal of Computational Physics

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“…A number of works has focused on structural optimization with either random Lamé parameters or forcing [3,10,11,13,36]. Stochastic models have also handled uncertainty in the geometry of the domain [7,25,33]. To ensure well-posedness of the stochastic problem, either an order must be defined on the relevant random variables, as in [11], or the problem needs to be transformed to a deterministic one by means of a probability measure.…”

Section: Introductionmentioning

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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Geometric Aspects of Shape Optimization

Плотников

Sokołowski

2023

J Geom Anal

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We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).

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The second order perturbation approach for elliptic partial differential equations on random domains

Cited by 10 publications

References 22 publications

Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Stochastic approximation for optimization in shape spaces

Geometric Aspects of Shape Optimization

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