GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Solving Complex-Valued Linear Systems via Equivalent Real Formulations
SIAM Journal on Scientific Computing
A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations
SIAM Journal on Scientific Computing
Separation-of-variables as a preconditioner for an iterative Helmholtz solver
Applied Numerical Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
On a class of preconditioners for solving the Helmholtz equation
Applied Numerical Mathematics
Component-Averaged Row Projections: A Robust, Block-Parallel Scheme for Sparse Linear Systems
SIAM Journal on Scientific Computing
A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems
SIAM Journal on Scientific Computing
A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation
Journal of Computational Physics
ACM Transactions on Mathematical Software (TOMS)
International Journal of Computer Mathematics - Fast Iterative and Preconditioning Methods for Linear and Non-Linear Systems
SIAM Journal on Scientific Computing
Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media
SIAM Journal on Scientific Computing
Preconditioning Helmholtz linear systems
Applied Numerical Mathematics
Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients
Journal of Computational and Applied Mathematics
GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
Numerical solution of the Helmholtz equation is a challenging computational task, particularly when the wave number is large. For two-dimensional problems, direct methods provide satisfactory solutions, but large three-dimensional problems become unmanageable. In this work, the block-parallel CARP-CG algorithm [Parallel Computing 36, 2010] is applied to the Helmholtz equation with large wave numbers. The effectiveness of this algorithm is shown both theoretically and practically, with numerical experiments on two- and three-dimensional domains, including heterogeneous cases, and a wide range of wave numbers. A second-order finite difference discretization scheme is used, leading to a complex, nonsymmetric and indefinite linear system. CARP-CG is both robust and efficient on the tested problems. On a fixed grid, its scalability improves as the wave number increases. Additionally, when the number of grid points per wavelength is fixed, the number of iterations increases linearly with the wave number. Convergence rates for heterogeneous cases are similar to those of homogeneous cases. CARP-CG also outperforms, at all wave numbers, one of the leading methods, based on the shifted Laplacian preconditioner with a complex shift and solved with a multigrid.