Preconditioned conjugate gradient

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

Nakano A (1997) Parallel multilevel preconditioned conjugate-gradient approach to variable-charge molecular dynamics. Comput Phys Commun 104(l-3) 59-69... 

BUSTER/TNT (Bricogne and Irvin, 1996) is another likelihood based refinement package that excels especially in cases in which the model is still severely incomplete (Blanc et al., 2004 Tronrud et al., 1987). It uses atomic parameters but also has a novel solvent and missing model envelope fimc-tion. The optimization method is a preconditioned conjugate gradient as implemented in the TNT package (Tronrud et al., 1987) that had a faithful audience in the pre-likelihood era. 

The recurrence relations for the preconditioned conjugate gradient (PCG) method can be derived from Algorithm [A2] after substituting x = M 1/2x and r + M1/2r. New search vectors d = M 1/2d can be used to derive the iteration process, and then the tilde modifiers dropped. The PCG method becomes the following iterative process. 

O. Axelsson and G. Lindskog, Numer. Math., 48, 449 (1989). On the Rate of Convergence of the Preconditioned Conjugate Gradient Method. 

Algorithm 4.2 Preconditioned Conjugate Gradient (PCC) for symmetric system solution or quadratic function minimization... 

Conjugate Gradient (CG) or Preconditioned Conjugate Gradient (PCG) methods (Chapter 6), which can become unstable with nonpositive definite matrices. 

Eisenstat, S.C., 1981, Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM J. Sci. Stat. Comput. 2, 1-4... 

Kincaid, D. R., Oppe, T. C., and Joubert, W. D., An Overview of NSPCG A Non-symmetric Preconditioned Conjugate Gradient Package , Center for Numerical Analysis Tech. Report CNA-228, Austin, Texas (1988). 

In terms of the computational work per outer Newton step k), TN methods based on preconditioned conjugate gradient require a Hessian-vector product (Hd) at each inner loop iteration, and one solution of a linear system Mz = r where M is the preconditioner. Since M may be sparse, this linear solution often takes a very small percentage of the total CPU time (e.g., <3% ). The benefits of faster convergence generally far outweigh these costs. 

Ghosh D, Avery P, Farhat C (2009) FETI-preconditioned conjugate gradient method for large-scale stochastic finite element problems. Int J Numer Methods Eng 80(6-7) 914-931... 

---