Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets
Computational Optimization and Applications
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Fast iterative solution of elliptic control problems in wavelet discretization
Journal of Computational and Applied Mathematics
ACM Transactions on Mathematical Software (TOMS)
Fast Conjugate Gradients with Multiple GPUs
ICCS '09 Proceedings of the 9th International Conference on Computational Science: Part I
Fault resilience of the algebraic multi-grid solver
Proceedings of the 26th ACM international conference on Supercomputing
The Chaotic Nature of Faster Gradient Descent Methods
Journal of Scientific Computing
Self-stabilizing iterative solvers
ScalA '13 Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems
Multiphysics simulations: Challenges and opportunities
International Journal of High Performance Computing Applications
Flexible global generalized Hessenberg methods for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
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An important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner-outer iterations [G. H. Golub and M. L. Overton, Numerical Analysis, Lecture Notes in Math. 912, Springer, Berlin, New York, 1982], where the preconditioner is solved by an inner iteration to a prescribed precision. In this paper, we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive definite system and analyze its convergence property. We establish a linear convergence result using a local relation of residual norms. We also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to high accuracy. The analysis is in agreement with observed numerical behavior of the algorithm. In particular, it suggests a heuristic choice of the stopping threshold for the inner iteration. Numerical examples are given to show the effectiveness of this choice and to compare the convergence bound.