Parallel preconditioned conjugate gradient algorithm on gpu. Matlab preconditioned conjugate gradient on big matrix. How to solve this system with conjugate gradient algorithm in. A generalization of the conjugategradient method to solve.
The computer code and data files made available on this web page are distributed under the gnu lgpl license. The conjugate gradient method aims to solve a system of linear equations, axb, where a is symmetric, without calculation of the inverse of a. Conjugate gradient method stanford engineering everywhere. Software downloads center for optimization and statistical. A vast range of explicit and implicit sparse preconditioners are covered, including the conjugate gradient, multilevel and fast multipole methods, matrix and operator splitting, fast fourier and wavelet transforms, incomplete lu and domain decomposition, schur complements and approximate inverses. If pcg fails to converge after the maximum number of iterations or halts for any reason, it displays a diagnostic message that includes the relative residual normbaxnormb and the.
Such preconditioners may be practically very efficient, however, their behavior is hard to predict theoretically. Oct 22, 2009 % this example demonstrates the use of conjgrad. The algorithm uses cocg conjugate orthogonal conjugate gradient or cocr conjugate orthogonal conjugate residual iterative solver preconditioned with the preconditioners proposed in the preprint. Solve system of linear equations conjugate gradients.
Conjugate gradient type methods and preconditioning. Conjugate gradient cg solver for linear systems cg is a fortran90 library which implements a simple version of the conjugate gradient cg method for solving a system of linear equations of the form axb, suitable for situations in which the matrix a is positive definite only real, positive eigenvalues and symmetric. Matrix preconditioning techniques and applications. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Preconditioning complex symmetric linear systems, by enrico bertolazzi and marco frego mathematical problems in engineering, volume 2015 2015. While preconditioned conjugate gradient methods work well for symmetric, positivedefinite matrices, other methods are necessary to treat large. It is faster than other approach such as gaussian elimination if a is wellconditioned. The conjugate gradient method has gained favour recently, notably as a procedure for solving large, preconditioned systems of algebraic equations. Conjugate gradient optimizer file exchange matlab central. Pdf nonsymmetric preconditioning for conjugate gradient. A multipreconditioned conjugate gradient algorithm request pdf. You could try to allow more fillin and thus try to improve the preconditioner by using a drop tolerance, e. It can be freely used for research, education or commercial purposes.
Week 8 introduction to numerical methods mathematics. Cg conjugate gradient cg solver for linear systems. Apparatus for optimizing the operational state of a system, which follows the karmarkar method, and which in the course of carrying out this method, obtains a solution to the linear system of equations ad 2 a t up in accordance with a modified conjugate gradient method that incorporates a preconditioning operation. Pdf a preconditioned conjugate gradient algorithm for generank. The following matlab project contains the source code and matlab examples used for conjugate gradient method. Nonsymmetric preconditioning for conjugate gradient and. Conjugate gradient method projects and source code download. Another popular preconditioner is m hht, where h is close to l. Lobpcg solves hermitian partial generalized eigenproblems using preconditioning, competes with eigs in matlab. We then of n are being very large, say, n 106 or n 107. Locally optimal block preconditioned conjugate gradient. An introduction to the conjugate gradient method without the agonizing pain edition 11 4 jonathan richard shewchuk august 4, 1994 school of computer science carnegie mellon university pittsburgh, pa 152 abstract the conjugate gradient method is the most prominent iterative method for solving sparse systems of linear equations. The conjugate gradient cg algorithm is one of the best known iterative. Robust preconditioned conjugate gradient for the gpu and parallel implementations rohit gupta, martin van gijzen, kees vuik gpu technology conference 2012, san jose ca.
The result is conjugate gradient on the normal equations cgnr. When the attempt is successful, cgs displays a message to confirm convergence. Your photo depicts an optimization problem rather than a system of equations, in which case you should be forming some square system of equations related to the kkt conditions and then solving it, possibly using a preconditioned conjugate gradient method. The conjugate gradient method can be applied to an arbitrary nbym matrix by applying it to normal equations a t a and righthand side vector a t b, since a t a is a symmetric positivesemidefinite matrix for any a. Solving a twodimensional elliptic model problem with the.
The algorithm uses cocg conjugate orthogonal conjugate gradient or cocr conjugate orthogonal conjugate residual iterative solver preconditioned. The technique of preconditioned conjugate gradient method consists in introducing a matrix c subsidiary. So what i am basically looking at is the iccg algorithm. R canonical tensor approximation, numerical linear algebra with applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The conjugate gradient method can be used to solve many large linear geophysical problems for example, leastsquares parabolic and hyperbolic radon transform, traveltime tomography, leastsquares migration, and fullwaveform inversion fwi e. If the null space na of a is explicitly known, then this divergence can be avoided by subtracting from b its orthogonal projection onto na. The developed optimized algorithm was tested repeatedly on a two dimensional poisson problem to ensure that it produced the same numerical solution as the original matlab function.
Indeed, spectral condition number of such matrices is too high. Preconditioned conjugate gradients, radial basis functions. Bicg solves not only the original linear system a x b but also the conjugate system a t x b. Preconditioned conjugate gradient methods for largescale. Mathematically equivalent to applying cg to normal equations atax. Several other algorithms can operate on symmetric positive definite matrices, but pcg is the quickest and most reliable at solving those types of systems 1. Henricus bouwmeester1, andrew dougherty1, and andrew v. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the. Read a nonlinearly preconditioned conjugate gradient algorithm for rank. Locally optimal block preconditioned conjugate gradient method, siam journal on scientific computing 23 2001, no. Cg is a matlab library which implements a simple version of the conjugate gradient cg method for solving a system of linear equations of the form axb, suitable for situations in which the matrix a is positive definite only real, positive eigenvalues and symmetric licensing.
It only requires a very small amount of membory, hence is particularly suitable for large scale systems. Matrix preconditioning techniques and applications ke. In this paper the preconditioned conjugate gradient method is used to solve the system of linear. A nonlinearly preconditioned conjugate gradient algorithm. Jun, 2019 this main function lobpcg is a version of the preconditioned conjugate gradient method algorithm 5. Conjugate gradient matlab code download free open source. In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positivedefinite. Conjugate gradients method for solving a system of linear equations ax f. The preconditioned conjugate gradients method pcg was developed to exploit the structure of symmetric positive definite matrices. In this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b, where a is a singular symmetric positive semidefinite matrix.
This text presents the theory and application of preconditioning techniques. The method diverges if b is not exactly in the range ra of a. A multipreconditioned conjugate gradient algorithm. The developed optimized algorithm was tested repeatedly on a two dimensional poisson problem to ensure that it produced the same numerical solution as. Locally optimal block preconditioned conjugate gradient file. Nonsymmetric preconditioning for conjugate gradient and steepest descent methods1 article pdf available in procedia computer science 511 december 2015 with 52 reads how we measure reads. Jul 25, 2006 solving linear systems resulting from the finite differences method or of the finite elements shows the limits of the conjugate gradient. The number of iterations required to achieve a given tolerance is independent of the number of variables. Preconditioned conjugate gradient algorithm november 21, 2016 in order to accelerate the iterative method, we usually introduce the socalled preconditioner bto the linear system ax b. The matlab function ichol computes by default the zero fillin variant of the incomplete cholesky factorisation. Conjugate gradient method file exchange matlab central. This leads to two sets of conjugate residuals defined in terms of the transpose of the coefficient matrix. Steepest descent and conjugate gradient methods with variable preconditioning authors. Topics covered include explicit and implicit sparse preconditioners, conjugate gradient, matrix and operator splitting, fast fourier and wavelet transforms, and schur complements.
The idea is that matrix aitself maybe illconditioned condition number of ais quite large, however, hopefully, by choose some good precondi. Knyazev, ilya lashuk submitted on 30 may 2006 v1, last revised 2 apr 2007 this version, v3. In this section we recall the main results from 19 and give some algorithmic details concerning the preconditioned conjugate gradient iteration. If cgs fails to converge after the maximum number of iterations or halts for any reason, it displays a diagnostic message that includes the relative residual normbaxnormb and the iteration. Based on your location, we recommend that you select. Conjugate gradient method projects and source code. Knyazev, toward the optimal preconditioned eigensolver. This matlab function attempts to solve the system of linear equations ax b for x using the preconditioned conjugate gradients method. The following matlab project contains the source code and matlab examples used for conjugate gradient.
This method is referred to as incomplete cholesky factorization see the book by golub and van loan for more details. Schieter tno, dg v institute of applied geoscience, 2600 ag delft, the netherlands received 8 february 1988 revised 18 june 1988 abstract. Preconditioned conjugate gradients for solving singular. Solve system of linear equations preconditioned conjugate. Pcg preconditioned conjugate gradients method is a replacement of the standard builtin pcg code. We use matlab to approximate the condition number a, a. When the attempt is successful, pcg displays a message to confirm convergence. Journal of computational and applied mathematics 24 1988 265275 265 northholland preconditioned conjugate gradients for solving singular systems e. T or m ttt is called preconditioner in naive implementation, each iteration requires multiplies by t and tt and a. An introduction to the conjugate gradient method without the. Nonsymmetric preconditioning for conjugate gradient and steepest descent methods 1. Preconditionnate conjugate gradient file exchange matlab.
The preconditioned conjugate gradient pcg has proven its efficiency and robustness in a wide range of applications. The package preqn automatically generates preconditioners when solving a sequence of linear systems of equations by means of the conjugate gradient method. The eigen library as i understand allows integration of the preconditioner to the conjugate gradient solver. Thus, our method applies to the corresponding toeplitz matrices. I am aiming to solve the linear equation ax b using the conjugate gradient technique with an incomplete cholesky preconditioner, leveraging the eigen library.
The biconjugate gradients bicg algorithm was developed to generalize the conjugate gradient cg method to nonsymmetric systems. Matrix preconditioning techniques and applications matlab. Usually, the matrix is also sparse mostly zeros and cholesky factorization is not feasible. Learn more preconditioner for block conjugate gradient in matlab. The simulation of largescale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. Conjugate gradient method for least squares cgls need. The existing preconditioned conjugate gradient method in matlab can be optimized in terms of wall clock time and, more importantly, required storage space. It is a multipreconditioned conjugate gradient algorithm mpcg 18 where each term in the definition of the preconditioners is considered to be a separate preconditioner. Preconditioning techniques have been developed for preparing the system for efficient solution by conjugate gradients. This main function lobpcg is a version of the preconditioned conjugate gradient method algorithm 5. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
1381 729 456 212 504 1313 871 1043 1497 1203 983 169 689 1218 908 524 686 278 1203 787 881 300 1022 732 707 1365 210 1442 920 701 166