Goal-oriented error estimation and adaptivity for the finite element method

Oden, J. T.; Prudhomme, Serge

doi:10.1016/s0898-1221(00)00317-5

Cited by 324 publications

(291 citation statements)

References 11 publications

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“…That data f O is assumed to be piecewise polynomial on subdomains of Ω and g O is assumed to be piecewise polynomial on subdomains of Γ N . One of the key ingredients in developing strategies to compute bounds for the output s is the definition of an auxiliary problem, denoted adjoint problem [1,10,16,14,17]. The variational form of the adjoint problem consists of finding ψ ∈ V such that…”

Section: Goal Oriented Simulations: Outputs and Adjoint Problemmentioning

confidence: 99%

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Numerical Meth Engineering

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SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

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Section: Goal Oriented Simulations: Outputs and Adjoint Problemmentioning

confidence: 99%

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Numerical Meth Engineering

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“…Prior to treatment, the LBIE Mesher 1 uses MRI data to generate a finite element mesh of the patient-specific biological domain. Goal-oriented estimation and adaption is used to optimize the mesh to a particular quantity of interest [9]. The tool then proceeds to solve an optimal control problem, wherein the laser parameters (location of optical fiber, laser power, etc.)…”

Section: Software Architecturementioning

confidence: 99%

Using Cyber-Infrastructure for Dynamic Data Driven Laser Treatment of Cancer

Bajaj

Oden

Diller

et al. 2007

Computational Science – ICCS 2007

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Abstract. Hyperthermia based cancer treatments are used to increase the susceptibility of cancerous tissue to subsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher intensity heat sources may be used to ablate the tissue. Utilizing the guidance of real-time treatment data while applying a laser heat source has the potential to provide unprecedented control over the outcome of the treatment process [6,12]. The goals of this work are to provide a working snapshot of the current system architecture developed to provide a real-time finite element solution of the problems of calibration, optimal heat source control, and goal-oriented error estimation applied the equations of bioheat transfer and demonstrate that current finite element technology, parallel computer architecture, peer-to-peer data transfer infrastructure, and thermal imaging modalities are capable of inducing a precise computer controlled temperature field within the biological domain.

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“…A general theory for controling this type of modeling error in quantities of interest Q through a posteriori error estimation and adaptive modeling has been developed by Oden and Prudhomme [19,17], and Oden and Vemaganti [15,16]. Techniques for deriving a posteriori error estimates for ε approx and goal-oriented adaptive meshing have been advanced by Babuska and Strouboulis [3], Oden and Prudhomme [18,19], Rannacher, Becker and others [9].…”

Section: Coarse and Discrete Modelsmentioning

confidence: 99%

“…ε mod ≈ 1 2 (γ low mod + γ upp mod )). If these estimated errors exceed preset tolerances, the errors must be reduced by adaptive meshing (for ε approx ) (see [17]) and adaptive modeling (for ε mod ).…”

Section: Coarse and Discrete Modelsmentioning

confidence: 99%

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A Computational Infrastructure for Reliable Computer Simulations

Oden

Browne

Babuška

et al. 2003

Lecture Notes in Computer Science

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Abstract. The reliability of computer simulations of physical events or of engineering systems has emerged as the single most critical issue facing advancements in computational engineering and science. Without concrete and quantifiable measures of reliability, the confidence and usefulness of computer predictions are severely limited and the value of computer simulation, in general, is greatly diminished. This paper describes a mathematical and computational infrastructure for the systematic validation of computer simulations of complex physical systems and presents procedures for the integration of computer-based verification and validation processes into simulations. A general class of applications characterized by variational boundary-value problems in continuum mechanics is considered, but the approach is valid for virtually all types of models used in simulations. To simulate a particular feature or attribute of a physical event, four basic concepts are used: 1) hierarchical modeling and a posteriori estimation of modeling error, 2) a posteriori error estimation of approximation error, 3) quantification of uncertainty in the data and in the predicted response, and 4) the development of dynamic a data management paradigm, based on code composition of associated code interfaces and use of annotated source code. The result is a computational framework for computing bounds on, and estimating accuracy of, computer predictions of user-specified features of the response of physical systems.

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Goal-oriented error estimation and adaptivity for the finite element method

Cited by 324 publications

References 11 publications

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Using Cyber-Infrastructure for Dynamic Data Driven Laser Treatment of Cancer

A Computational Infrastructure for Reliable Computer Simulations

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