The construction of preconditioners for elliptic problems by substructuring. I
Mathematics of Computation
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Preconditioned conjugate gradients for solving singular systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
On the conjugate gradient solution of the Schur complement system obtained from domain decomposition
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific and Statistical Computing
A Restarted GMRES Method Augmented with Eigenvectors
SIAM Journal on Matrix Analysis and Applications
Balancing domain decomposition for problems with large jumps in coefficients
Mathematics of Computation
Matrix computations (3rd ed.)
Journal of Computational Physics
A Deflated Version of the Conjugate Gradient Algorithm
SIAM Journal on Scientific Computing
Analysis of acceleration strategies for restarted minimal residual methods
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
Multigrid
On the Construction of Deflation-Based Preconditioners
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow
SIAM Journal on Numerical Analysis
A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems
SIAM Journal on Scientific Computing
A Comparison of Deflation and the Balancing Preconditioner
SIAM Journal on Scientific Computing
On deflation and singular symmetric positive semi-definite matrices
Journal of Computational and Applied Mathematics
Additive and Multiplicative Two-Level Spectral Preconditioning for General Linear Systems
SIAM Journal on Scientific Computing
Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems
SIAM Journal on Scientific Computing
Fast and robust solvers for pressure-correction in bubbly flow problems
Journal of Computational Physics
A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation
SIAM Journal on Matrix Analysis and Applications
A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps
SIAM Journal on Scientific Computing
A parallel two-level preconditioner for cosmic microwave background map-making
SC '12 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Journal of Computational Physics
Scalable domain decomposition preconditioners for heterogeneous elliptic problems
SC '13 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Journal of Computational and Applied Mathematics
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For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.