Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning

Knyazev, Andrew; Lashuk, Ilya

doi:10.1137/060675290

Cited by 54 publications

(53 citation statements)

References 11 publications

(30 reference statements)

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“…Our optimization chain is driven by a gradient‐based optimization algorithm, in particular the CG algorithm . The gradients

\frac{d J}{d boldD}

required, that is, the sensitivities of the objective function J to changes in the design variables D , could be approximated using finite differences.…”

Section: Gradients Via the Adjoint Approachmentioning

confidence: 99%

Adjoint‐based airfoil optimization with discretization error control

Hartmann

2014

Numerical Methods in Fluids

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SUMMARYIn this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Our optimization chain is driven by a gradient‐based optimization algorithm, in particular the CG algorithm . The gradients

\frac{d J}{d boldD}

required, that is, the sensitivities of the objective function J to changes in the design variables D , could be approximated using finite differences.…”

Section: Gradients Via the Adjoint Approachmentioning

confidence: 99%

Adjoint‐based airfoil optimization with discretization error control

Hartmann

2014

Numerical Methods in Fluids

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“…To obtain the biomass parameters we have fitted the shifted Gaussian to the measured chlorophyll profiles by adjusting the parameter values. The conjugate gradient method was used (Baldick, 2006;Knyazev and Lashuk, 2008). The shifted Gaussian was convergent for each chlorophyll profile.…”

Section: Testing the Shifted Gaussian Solutionmentioning

confidence: 99%

Extended Formulations and Analytic Solutions for Watercolumn Production Integrals

Kovač

Platt

Antunović³

et al. 2017

Front. Mar. Sci.

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The effect of biomass dynamics on the estimation of watercolumn primary production is analyzed, by coupling a primary production model to a simple growth equation for phytoplankton. The production model is formulated with depth-and time-resolved biomass, and placed in the context of earlier models, with emphasis on the canonical solution for watercolumn production. A relation between the canonical solution and the general solution for the case of an arbitrary depth-dependent biomass profile was derived, together with an analytical solution for watercolumn production in case of a depth dependent biomass profile described with the shifted Gaussian function. The analysis was further extended to the case of a time-dependent, mixed-layer biomass, and two additional analytical solutions to this problem were derived, the first in case of increasing mixed-layer biomass and the second in case of declining biomass. The solutions were tested with Hawaii Ocean Time-series data. The canonical solution for mixed-layer production has proven to be a good model for this data set. The shifted Gaussian function was demonstrated to be an accurate model for the measured biomass profiles and the shifted Gaussian parameters extracted from the measured profiles were further used in the analytical solution for watercolumn production and results compared with data. The influence of time-dependent biomass on mixed-layer production was studied through analytical solutions. Re-examining the Critical Depth Hypothesis we derived an expression for the daily increase in mixed-layer biomass. Finally, the work was placed in a remote sensing context and the time-dependent model for biomass related to the remotely sensed-biomass.

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“…There are very efficient algorithms for both steps. The powerful iterative techniques such as the conjugate gradient method may be applied to solve the equations for finding pressure distribution. The Hoshen and Kopelman algorithm can be used to recognize the clusters.…”

Section: Resultsmentioning

confidence: 99%

Effect of connectivity misrepresentation on accuracy of upscaled models in oil recovery by CO₂ injection

2015

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An upscaling method such as renormalization converts a detailed geological model to a coarse one. Although flow equations can be solved faster on a coarse model, its results have more errors. Numerical dispersion, heterogeneity loss, and connectivity misrepresentation are the factors responsible for errors. Connectivity has a great effect on the fluid distribution and leakage pathways in EOR processes or CO2 storage. This paper deals with the description and quantification of connectivity misrepresentation in the upscaling process. For detection of high‐flow regions, the flow equations are solved under simplified single‐phase conditions. These regions are recognized as the cells whose fluxes are greater than a cut‐off. The connectivity is investigated by checking whether the high‐flux region forms a spanning cluster across the model. An indicator called the flux connectivity distortion is defined to measure the connectivity misrepresentation of coarse grids. This indicator is shown to have a strong correlation with the oil saturation error. For renormalization method, this relation helps to propose an upper limit of coarsening beyond which the coarse model results are not reliable. Once the proper sizes of coarse blocks are determined, flow behavior can be simulated accurately in a project of CO2‐EOR or CO2 sequestration. © 2015 Society of Chemical Industry and John Wiley & Sons, Ltd

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Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning

Cited by 54 publications

References 11 publications

Adjoint‐based airfoil optimization with discretization error control

Adjoint‐based airfoil optimization with discretization error control

Extended Formulations and Analytic Solutions for Watercolumn Production Integrals

Effect of connectivity misrepresentation on accuracy of upscaled models in oil recovery by CO₂ injection

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Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning

Cited by 54 publications

References 11 publications

Adjoint‐based airfoil optimization with discretization error control

Adjoint‐based airfoil optimization with discretization error control

Extended Formulations and Analytic Solutions for Watercolumn Production Integrals

Effect of connectivity misrepresentation on accuracy of upscaled models in oil recovery by CO2 injection

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Product

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Effect of connectivity misrepresentation on accuracy of upscaled models in oil recovery by CO₂ injection