A Goal-Oriented Error Estimator for a Class of Homogenization Problems

Carraro, Thomas; Goll, Christian

doi:10.1007/s10915-016-0338-y

“…It becomes an estimation after substituting the unknown continuous solutions with computable quantities. As shown in several applications [9,30,29,26,10,5] one efficient method to produce computable quantities is the use of a patch-wise higher-order approximation of the terms z h and w h . In particular, we have used the following interpolation operator with…”

Section: A Posteriori Error Estimatorsmentioning

confidence: 99%

“…[2,33]. Furthermore, goal oriented a posteriori error estimation has been successfully used in many applications, see the seminal works [6,4] for an overview of the Dual Weighed Residual (DWR) technique and exemplarily [9,30,34,8] for some specific applications. A posteriori error estimation methods have been used to control the discretization error either in global norms, e.g.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

Carraro

¹

,

Dörsam

²

,

Frei

³

et al. 2018

J Optim Theory Appl

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In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first order optimality conditions by the Newton-type method. This strategy allows to balance the two errors and to derive effective stopping criteria for the Newton-iterations. The algorithm proceeds with the search of the optimal point on coarse grids which are refined only if the discretization error becomes dominant. Using computable error indicators the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.

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“…For a set of slightly different homogenised problems, [19] presents the goaloriented error estimation for general quantities of interest. We also point out the existing literature for goal-adaptivity in the context of contaminant transport, presented in the articles [10,39], but which differs slightly from the work presented here.…”

mentioning

confidence: 99%

Linearization of the Travel Time Functional in Porous Media Flows

Houston¹,

Rourke²,

Zee³

2022

SIAM J. Sci. Comput.

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The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gâteaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.

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“…For a set of slightly different homogenised problems, [20] presents the goal-oriented error estimation for general quantities of interest. We also point out the existing literature for goal-adaptivity in the context of contaminant transport, presented in the articles [11,38], but which differs slightly from the work presented here.…”

mentioning

confidence: 99%

Linearisation of the Travel Time Functional in Porous Media Flows

Houston¹,

Rourke²,

Zee³

2021

Preprint

0

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The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gâteaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.

show abstract

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

Cited by 4 publications

References 36 publications

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

Linearization of the Travel Time Functional in Porous Media Flows

Linearisation of the Travel Time Functional in Porous Media Flows

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