scholarly journals Comparative study of conjugate gradient algorithms performance on the example of steady-state axisymmetric heat transfer problem

2013 ◽  
Vol 34 (3) ◽  
pp. 15-44 ◽  
Author(s):  
Paweł Ocłoń ◽  
Stanisław Łopata ◽  
Marzena Nowak
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Conjugate Gradient ◽  
Transfer Problem ◽  
Heat Conduction Problem ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Conjugate Gradient Algorithms ◽  
Gradient Algorithms ◽  
Biconjugate Gradient

Abstract The finite element method (FEM) is one of the most frequently used numerical methods for finding the approximate discrete point solution of partial differential equations (PDE). In this method, linear or nonlinear systems of equations, comprised after numerical discretization, are solved to obtain the numerical solution of PDE. The conjugate gradient algorithms are efficient iterative solvers for the large sparse linear systems. In this paper the performance of different conjugate gradient algorithms: conjugate gradient algorithm (CG), biconjugate gradient algorithm (BICG), biconjugate gradient stabilized algorithm (BICGSTAB), conjugate gradient squared algorithm (CGS) and biconjugate gradient stabilized algorithm with l GMRES restarts (BICGSTAB(l)) is compared when solving the steady-state axisymmetric heat conduction problem. Different values of l parameter are studied. The engineering problem for which this comparison is made is the two-dimensional, axisymmetric heat conduction in a finned circular tube.

Implementation of Modified Conjugate Gradient Algorithm in Electromagnetic Tomography Lab System

2013 ◽  
Vol 655-657 ◽  
pp. 693-696
Author(s):  
Zhan Jun Wang ◽  
Liu Li
Keyword(s):  
Conjugate Gradient ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Reconstructed Image ◽  
Conjugate Gradient Algorithms ◽  
Flow Models ◽  
Electromagnetic Tomography ◽  
Tomography System ◽  
Gradient Algorithms

The advantage of the electromagnetic tomography is introduced briefly. Based on conjugate gradient algorithm, modified conjugate gradient algorithm for electromagnetic tomography (EMT) is proposed, which improves quality of reconstructed image and convergence speed efficiently. In the light of the lab electromagnetic tomography system, modified conjugate gradient for reconstructing images is verified. By evaluation of image error and the relevance, regularization, Landweber, conjugate gradient and modified conjugate gradient algorithms are compared. It can draw the conclusion that for different flow models, image error and the correlation using modified conjugate gradient algorithm is superior to the others in lab EMT system.

scholarly journals A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shashi Kant Mishra ◽  
Suvra Kanti Chakraborty ◽  
Mohammad Esmael Samei ◽  
Bhagwat Ram
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Test Problems ◽  
Conjugate Gradient Algorithms ◽  
Gradient Algorithms ◽  
Unconstrained Optimization Problems ◽  
Wolfe Conditions ◽  
Nonlinear Unconstrained Optimization

AbstractA Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1.

scholarly journals On the use of adjoints in the inversion of observed quasi-static deformation

2020 ◽  
Vol 224 (2) ◽  
pp. 896-908
Author(s):  
D W Vasco ◽  
Gwyn Mali
Keyword(s):  
Conjugate Gradient ◽  
Adjoint Operator ◽  
Central Valley ◽  
Gradient Algorithm ◽  
Gradient Estimates ◽  
Conjugate Gradient Algorithms ◽  
Reservoir Volume ◽  
Gradient Algorithms ◽  
Residual Displacements ◽  
Model Update

SUMMARY An adjoint-based conjugate gradient algorithm provides an efficient means for imaging sources of deformation within the Earth, such as volume stresses associated with fluid flow in aquifers and reservoirs. For time intervals over which the overburden deforms elastically, one can calculate the gradient elements for a single model update using just two numerical simulations. The first is a forward run that is used to compute the residuals associated with the given iteration. The second simulation is to evaluate the application of the adjoint operator to the residuals. In this adjoint calculation, the residual displacements are applied as sources at the measurement locations, driving the deformation in the simulation. The volume stress on the source grid blocks, in response to the residual displacements, provide the gradient components. We apply this technique to satellite-based interferometric synthetic aperture radar (InSAR) line-of-sight displacements that were observed over an oil reservoir in California’s Central Valley. We find that the adjoint-based gradient estimates, requiring 18 CPU seconds, agree with conventional numerical calculations that take over 3700 CPU seconds to compute. Conjugate gradient algorithms utilizing the conventional approach and adjoint-based gradient computations give roughly the same reductions in misfit and similar final estimates of reservoir volume change.

scholarly journals The analysis of gradient algorithm effectiveness - two dimensional heat transfer problem

2010 ◽  
Vol 31 (4) ◽  
pp. 37-50
Author(s):  
Stanisław Łopata ◽  
Paweł Ocłoń
Keyword(s):  
Heat Transfer ◽  
Transfer Problem ◽  
Relaxation Method ◽  
Computer Code ◽  
Gradient Algorithm ◽  
Heat Transfer Problem ◽  
Computational Time ◽  
Two Dimensional ◽  
Gradient Algorithms ◽  
Biconjugate Gradient

The analysis of gradient algorithm effectiveness - two dimensional heat transfer problemThe analysis of effectiveness of the gradient algorithm for the two-dimension steady state heat transfer problems is being performed. The three gradient algorithms - the BCG (biconjugate gradient algorithm), the BICGSTAB (biconjugate gradient stabilized algorithm), and the CGS (conjugate gradient squared algorithm) are implemented in a computer code. Because the first type boundary conditions are imposed, it is possible to compare the results with the analytical solution. Computations are carried out for different numerical grid densities. Therefore it is possible to investigate how the grid density influences the efficiency of the gradient algorithms. The total computational time, residual drop and the iteration time for the gradient algorithms are additionally compared with the performance of the SOR (successive over-relaxation) method.

scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

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