Since it involves conjugate directions, it too terminates after m. An introduction to the conjugate gradient method without. An introduction to the conjugate gradient method without the. Reidon the method of conjugate gradients for the solution of large sparse. This removes the singularity caused by the primary potential, resulting in an accurate 3d resistivity model. Mathematically equivalent to applying cg to normal equations. Gpu implementation of splitting up conjugate gradient. Incomplete cholesky factorization preconditioned conjugate gradient iccg method is effective to solve large sparse symmetric positive definite linear systems.
A new implementation of the nonlinear conjugate gradient. The conjugate gradient method the conjugate gradient method cg is to be described in this section. Preconditioning 47 a notes 48 b canned algorithms 49 b1. 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. This routine can be used to solve symmetric systems as well as nonsymmetric. Let tp 0 be an initial guess vector for the solution. Outline of the nonlinear conjugate gradient method 42 14. The conjugate gradient method cg, an iterative method of solving linear.
As a result of operation of this method we obtain a sequence of vectors starting from. This method will be called the conjugate gradient method or, more briefly, the cg method, for reasons which will unfold from the theory developed in later sections. A modified conjugate gradient method and its global. Compared with cg method that has no preprocessing, iccg method is faster and more stable method. Cgs method conjugate gradient squared cgs method matlab. Pcg2 uses the preconditioned conjugategradient method to solve the equations. A survey of the conjugate gradient method michael lam math 221 final project the conjugate gradient method cg was developed independently by hestenes and stiefel 1 back in the 1950s and has enjoyed widespread use as robust method for solving linear systems and eigenvalue problems when the associated matrix is symmetric and positive definite. Axelsson and lindskog 1986 presented a preconditioner that commonly is called the modified. An elementbyelement preconditioned conjugate gradient. Then we switch to the nonlinear conjugate gradient method.
The linear conjugate gradient method is discussed in details in numerical analysis ii, the second semester of this course. The conjugate gradient method was originally developed by hestenes and stiefelll. The conjugate gradient method is a conjugate direction method. Implementing conjugate gradients with incomplete cholesky. Comparison of steepest descent method and conjugate. They compare sp cg spbicg with iccg ilubicg and show that the sp method. Now are going to discuss it brie y the linear conjugate gradient methods in order to understand the logic behind its nonlinear versions. Siam journal on scientific and statistical computing. Stiefel in 12, 20 used the theory of orthogonal polynomials to discuss best iterative methods for solving finite systems of linear algebraic equations. An attractive feature of the new conjugate gradient. However, iccg method requires solving two sparse triangular linear systems during each iteration. Acceleration of linear finitedifference poissonboltzmann. On the robustness of conjugategradient methods and quasi. Conjugate gradient algorithms and finite element methods.
The conjugate directions are not specified beforehand, but rather. The cg method searches for the exact solution along a series of conjugate directions, and is implemented as an iterative procedure as follows. Methods of conjugate gradients for solving linear systems. A modified conjugate gradient method and its global convergence can li college of mathematics, honghe university, mengzi 661199, china abstract conjugate gradient method is an important method for solving largescale unconstrained nonlinear optimization problems. The iterative method is based on the conjugate gradient algorithm applied to the implicitly formed normal equations. Parallel iccg on a hierarchical memorymultiprocessor. This paper considers an efficient preconditioner and proposes a multigrid preconditioned conjugate gradient method mgcg method which is the conjugate gradient method. Although the speed of convergence of the iccg method is fast, it is difficult to parallelize the method straightforwardly, and thus several variants have been. The m iccg method seems to be one of the most efficient methods among the iterative methods. Therefore, the performance of conjugate gradient solvers with meshfree methods is investigated and issues arising from the nature of meshfree methods are noted. The gradient is rfu au b, so the steepest descent direction is given by the residual.
In iccg, the calculation speed of cg method is enhanced with preprocessing technology incomplete cholesky factorization. However, alternative methods of matrix preconditioning have been developed to achieve more efficient conjugate gradient solvers. The conjugate gradient method is a low storage algorithm that. Apparent resistivity can be computed for di erent sourcereceiver array con guration. Somersalo cgls method conjugate gradient method for least squares cgls need. The iccg0 method has been used in production in lasnex with matrices. In a paper by hestenes and stiefel 14, the conjugate gradient method was originally presented 19, p.
Introduction and new method the nonlinear conjugate gradient method is a wellknown method for largescale unconstrained optimization problems 1 min x2rn fx. An incomplete splittingup conjugate gradient method for parallel computing 237 iccg method is used for solving a symmetric positivedefinite matrix. But then rn 2rn is orthogonal to all vectors in rn so rn 0 and hence xn x. Indeed, if the solution is obtained in m gradient vanishes and the algorithm. If you see this picture, then weve not only made headway with the conjugate gradient method, which is a big deal for solving linear systems, but also weve made headway with the conjugate gradient method for minimizing function.
The incomplete choleskyconjugate gradient method for the. T or m ttt is called preconditioner in naive implementation, each iteration requires multiplies. According to the following paper, the speed of convergence of spcg is comparable to that of iccg. Abstract the conjugate gradient method cg, an iterative method of. Although the steepest descent method converges, it is inef. The method uses a preconditioner to accerlerate convergence. Eigenvectors are explained and used to examine the convergence of the jacobi method. Pdf we investigate the effect of the ordering of the unknowns on the convergence of the. By decomposing the matrix and substituting back to, we can compute the electric potential these values need to be stored to formulate the jacobimatrixininversionateverynodecorrespondingtoall point electric sources. Via their combinations practitioners have been able to solve complicated, direct and inverse, multidemensional problems. An incomplete splittingup conjugate gradient method for. Cgn the method of conjugate gradients for normal equations.
The conjugate gradient method is a successful iterative method see 5, section 10. In addition, alternative preconditioning methods in conjunction with a conjugate gradient solver are implemented and tested. Application of preconditioned conjugate gradient method to. Iccglu a fortran iv subroutine to solve large sparse. The conjugate gradient method is valued in that it suits to parallel computing and even illconditioned problems can be easily solved with the help of a good preconditioning. Selects the successive direction vectors as a conjugate version of the successive gradients obtained as the method progresses. The conjugate gradient method for linear and nonlinear.
We choose the direction vector d 0 to be the steepest descent directionof the function fu. The inherent serialism of solving sparse triangular becomes a bottleneck which prevents high efficient parallelization of iccg. In the method, the conjugate directions are chosen by using the gradients of the quadratic function and these directions are computed iteratively. We discuss only the computation involved in the iccg method. Pdf the effect of ordering on preconditioned conjugate gradient. This section establishes the multipreconditioned analogy of cg in a fashion similar to the derivation of the standard pcg, whose. We use conjugate gradient method to solve the system of linear equations given in the form of ax b. The preconditioners considered are the incomplete cholesky iccg and the. The incomplete cholesky preconditioner iccg has been very popular. Pdf efficient iccg on a shared memory multiprocessor. In theory, the successive gradients generated by the. And if the function wasnt quadratic, and our equations werent linear, the conjugate gradient idea would still be. Performance of parallel conjugate gradient solvers in. View the article pdf and any associated supplements and figures for a period of 48 hours.
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