A new goal-oriented formulation of the finite element method

Kenan, Kergrene; Prudhomme, Serge; Chamoin, Ludovic; Laforest, Marc

doi:10.1016/j.cma.2017.09.018

Cited by 25 publications

(23 citation statements)

References 15 publications

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“…The idea of goal-oriented adaptive (finite element) methods date back to the late 1990s, see e.g., [1,30,28] for early works and analysis, and [13,21,38,11,16] for some recent new developments. These methods are based on a different idea than the machine-learning framework that we propose.…”

Section: Related Literaturementioning

confidence: 99%

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A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

Brevis

Muga

Zee

2021

Computers & Mathematics with Applications

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Section: Related Literaturementioning

confidence: 99%

“…This optimal test function is also obtained in our framework via (21), by using a limiting weight of the form:…”

Section: D Diffusion With One Qoimentioning

confidence: 99%

A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

Brevis

Muga

Zee

2021

Computers & Mathematics with Applications

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“…where we introduce the local displacement output obtained from the compatible solution  k (u k ), noting that a similar expression can be obtained from the equilibrated solution which is denoted as  k ( s ). Now we manipulate Equation (23) to force a term that can be bounded to appear. We do so by adding and subtracting…”

Section: Local Outputs and Their Boundsmentioning

confidence: 99%

“…There are several works that takes the concept of goal-oriented formulations and apply it to reduced order methods, many specific for the PGD, but few that explore the use of the dual analysis in this context. [21][22][23][24] The solutions obtained from dual analysis are ideal when working with QoI, as they can provide high quality bounds of error on the QoI of the worst solution. 25 Dual analysis is also effective to drive adaptivity mesh refinement processes, as it provides information of the errors in the elements either in a global or in a local framework.…”

Section: Introductionmentioning

confidence: 99%

Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Reis

Almeida

Dı́ez

et al. 2020

Numerical Meth Engineering

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When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable solutions, so that a large number of design options can be assessed. In this context, the proper generalized decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the “curse of dimensionality.” Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking quantities of interest with guaranteed bounds. In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain quantities of interest and their bounds, which are guaranteed. We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those obtained using a uniform refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest.

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“…The dual weighted residual (DWR) method promoted by Becker and Rannacher [5], see also [3,7,22,24,33], the general framework by Prodhumme and Oden [36,40], the approach of Maday and Patera [31], multi-objective error estimation in [16,23,46], enhanced least-squares finite element methods by Chaudhry et al [8], or the constitutive relation error (CRE) approach of Ladevèze et al [28,30] and Rey et al [42][43][44], see also the references therein, are very popular approaches to goal-oriented error estimation; this can also be built in the discretization scheme as in Kergrene et al [27]. The obtained bounds are, however, often not guaranteed in the sense of yielding a fully computable number that is rigorously greater than or equal to the goal error.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

Mallik

Vohralı́k

Yousef

2020

Journal of Computational and Applied Mathematics

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We derive a unified framework for goal-oriented a posteriori estimation covering in particular higher-order conforming, nonconforming, and discontinuous Galerkin finite element methods, as well as the finite volume method. The considered problem is a model linear second-order elliptic equation with inhomogeneous Dirichlet and Neumann boundary conditions and the quantity of interest is given by an arbitrary functional composed of a volumetric weighted mean value (source) term and a surface weighted mean (Dirichlet boundary) flux term. We specifically do not request the primal and dual discrete problems to be resolved exactly, allowing for inexact solves. Our estimates are based on H(div)-conforming flux reconstructions and H 1-conforming potential reconstructions and provide a guaranteed upper bound on the goal error. The overall estimator is split into components corresponding to the primal and dual discretization and algebraic errors, which are then used to prescribe efficient stopping criteria for the employed iterative algebraic solvers. Numerical experiments are performed for the finite volume method applied to the Darcy porous media flow problem in two and three space dimensions. They show excellent effectivity indices even in presence of primal and dual algebraic errors and enable to spare a large percentage of unnecessary algebraic iterations.

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A new goal-oriented formulation of the finite element method

Cited by 25 publications

References 15 publications

A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

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