On Optimized Extrapolation Method for Elliptic Problems with Large Coefficient Variation

Garbey, Marc; Shyy, Wei

doi:10.1260/174830107783133851

Cited by 4 publications

(2 citation statements)

References 31 publications

(66 reference statements)

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“…But, in the case of stiff problems we would like to get some adaptivity on the construction of the weight function. This is an open problem that we are currently working on [13]. After exposing the theoretical concepts behind the optimized extrapolation method, we are now going to present different numerical illustrations of this work.…”

Section: Task 2: Optimized Extrapolationmentioning

confidence: 99%

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

Garbey

Picard

2008

Acta Mech Sin

Self Cite

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The goal of this paper is to present a versatile framework for solution verification of PDE's. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W. Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are separately an incompressible back step flow test case and a heat transfer problem for a battery. Our error estimate might be ultimately verified with a near by manufactured solution. While our procedure is systematic and requires numerous computation of residuals, one can take advantage of distributed computing to get quickly the error estimate.

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Section: Task 2: Optimized Extrapolationmentioning

confidence: 99%

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

Garbey

Picard

2008

Acta Mech Sin

Self Cite

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show abstract

“…In the LSE method [18], [19] we chose the discrete L 2 norm. The choice of the L 1 or the L ∞ norm has been tested for stiff elliptic problems [20]. One of the difficulties encountered with a two-level extrapolation method is the existence of subsets of M (h ∞ ) whereŨ 1 andŨ 2 are much closer to each other than what the expected order of accuracy based on local error analysis should provide.…”

Section: A General Conceptmentioning

confidence: 99%

Toward a General Solution Verification Method for Complex PDE Problem with Hands Off Coding

Garbey¹,

Picard

2007

45th AIAA Aerospace Sciences Meeting and Exhibit

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The goal of this paper is to present a versatile framework for solution verification of PDEs. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W.Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are respectively an incompressible back step flow test case and a heat transfer problem for a battery. Both test cases are solved by the Finite Element package ADINA. Our error estimate might be ultimately verified with a near by manufactured solution. While our procedure is systematic and requires numerous computation of residuals, one can take advantage of grid computing to get quickly the error estimate.

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A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

Nachaoui

Afraites

Hadri

et al. 2022

CPAA

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<p style='text-indent:20px;'>This paper introduce a novel optimization procedure to reduce mixture of Gaussian and impulse noise from images. This technique exploits a non-convex PDE-constrained characterized by a fractional-order operator. The used non-convex term facilitated the impulse component approximation controlled by a spatial parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. A non-convex and non-smooth bi-level optimization framework with a modified projected gradient algorithm is then proposed in order to learn the parameter <inline-formula><tex-math id="M2">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. Denoising tests confirm that the non-convex term and learned parameter <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> lead in general to an improved reconstruction when compared to results of convex norm and manual parameter <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> choice.</p>

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On Optimized Extrapolation Method for Elliptic Problems with Large Coefficient Variation

Cited by 4 publications

References 31 publications

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

Toward a General Solution Verification Method for Complex PDE Problem with Hands Off Coding

A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

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