An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations
SIAM Journal on Scientific Computing
Parallelization of multilevel ILU preconditioners on distributed-memory multiprocessors
PARA'10 Proceedings of the 10th international conference on Applied Parallel and Scientific Computing - Volume Part I
Modification and Compensation Strategies for Threshold-based Incomplete Factorizations
SIAM Journal on Scientific Computing
Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers
Journal of Computational and Applied Mathematics
A rapidly converging domain decomposition method for the Helmholtz equation
Journal of Computational Physics
Preconditioning of complex symmetric linear systems with applications in optical tomography
Applied Numerical Mathematics
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An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete $LDL^T$ factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls $\|L^{-1}\|$ for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a “black-box” solver to a series of challenging two- and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.