A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations
SIAM Journal on Scientific and Statistical Computing
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Algebraic multilevel preconditioning methods, II
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific and Statistical Computing
A flexible inner-outer preconditioned GMRES algorithm
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
Efficient management of parallelism in object-oriented numerical software libraries
Modern software tools for scientific computing
Inner and Outer Iterations for the Chebyshev Algorithm
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Analysis of iterative methods for saddle point problems: a unified approach
Mathematics of Computation
Flexible Inner-Outer Krylov Subspace Methods
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
ICA3PP '02 Proceedings of the Fifth International Conference on Algorithms and Architectures for Parallel Processing
Minimizing communication in sparse matrix solvers
Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis
Enabling high-fidelity neutron transport simulations on petascale architectures
Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis
Composable Linear Solvers for Multiphysics
ISPDC '12 Proceedings of the 2012 11th International Symposium on Parallel and Distributed Computing
Multiphysics simulations: Challenges and opportunities
International Journal of High Performance Computing Applications
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The solution of large, sparse linear systems is often a dominant phase of computation for simulations based on partial differential equations, which are ubiquitous in scientific and engineering applications. While preconditioned Krylov methods are widely used and offer many advantages for solving sparse linear systems that do not have highly convergent, geometric multigrid solvers or specialized fast solvers, Krylov methods encounter well-known scaling difficulties for over 10,000 processor cores because each iteration requires at least one vector inner product, which in turn requires a global synchronization that scales poorly because of internode latency. To help overcome these difficulties, we have developed hierarchical Krylov methods and nested Krylov methods in the PETSc library that reduce the number of global inner products required across the entire system (where they are expensive), though freely allow vector inner products across smaller subsets of the entire system (where they are inexpensive) or use inner iterations that do not invoke vector inner products at all. Nested Krylov methods are a generalization of inner-outer iterative methods with two or more layers. Hierarchical Krylov methods are a generalization of block Jacobi and overlapping additive Schwarz methods, where each block itself is solved by Krylov methods on smaller blocks. Conceptually, the hierarchy can continue recursively to an arbitrary number of levels of smaller and smaller blocks. As a specific case, we introduce the hierarchical FGMRES method, or h-FGMRES, and we demonstrate the impact of two-level h-FGMRES with a variable preconditioner on the PFLOTRAN subsurface flow application. We also demonstrate the impact of nested FGMRES, BiCGStab and Chebyshev methods. These hierarchical Krylov methods and nested Krylov methods significantly reduced overall PFLOTRAN simulation time on the Cray XK6 when using 10,000 through 224,000 cores through the combined effects of reduced global synchronization due to fewer global inner products and stronger inner hierarchical or nested preconditioners.