The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.
Several other constructions of preconditioners @cite_10 @cite_4 @cite_28 are based on incomplete LU (ILU) decomposition, i.e., generating only a small portion of the entries of the LU factorization of the discrete Helmholtz operator and applying this ILU decomposition as a preconditioner. Recent approaches based on ILUT (incomplete LU factorization with thresholding) and ARMS (algebraic recursive multilevel solver) have been reported in @cite_28 . These ILU preconditioners bring down the number of iterations quite significantly, however the number of iterations still scale typically linearly in @math . In connection with the ILU preconditioners, the sweeping preconditioner can be viewed as an approximate LU (ALU) preconditioner: instead of keeping only a few selected entries, it approximates the whole inverse operator more accurately in a more sophisticated and effective form, thus resulting in substantially better convergence properties.
@cite_28 @cite_10 @cite_4
2063547741 2743824295 2098102734 2025033760
We present an incomplete LU preconditioner for solving discretized Helmholtz problems. The preconditioner is based on an analytic factorization of the Helmholtz operator. This allows us to take the physical properties of the acoustics problem modeled by the Helmholtz equation into account in the preconditioner. We show how the parameters in the preconditioner can be chosen in order to make it effective. Numerical experiments show that the new preconditioner leads to convergent iterative methods even for large wave numbers, and it outperforms classical ILU preconditioners by a large margin. Accelerating large-scale linear solvers is always crucial for scientific research and industrial applications. In this regard, preconditioners play a key role in improving the performance of iterative linear solvers. This paper presents a summary and review of our work about the development of parallel ILU preconditioners on GPUs. The mechanisms of ILU(0), ILU(k), ILUT, enhanced ILUT, and block-wise ILU(k) are reviewed and analyzed, which give a clear guidance in the development of iterative linear solvers. ILU(0) is the most commonly used preconditioner, and the nonzero pattern of its matrix is exactly the same as the original matrix to be solved. ILU(k) uses k levels to control the pattern of its preconditioner matrix. ILUT selects entries for its preconditioner matrix by setting thresholds without considering its original matrix pattern. In addition to point-wise ILU preconditioners, a block-wise ILU(k) preconditioner is designed delicately in support of block-wise matrices. In implementation, the RAS (Restricted Additive Schwarz) method is adopted to optimize the parallel structure of a preconditioner matrix. Coupling with the configuration parameters of ILU preconditioners, a complex situation appears in the parallel solution process, so decoupled algorithms are adopted. These algorithms are implemented and tested on NVIDIA GPUs. The experiment results show that a single-GPU implementation can speed up an ILU preconditioner by a factor of 10, compared to traditional CPU implementation. The results also show that the ILU(0) has better speedup than ILU(k) but slower convergence than ILU(k). Level k of ILU(k) and threshold (p, t) of ILUT are effective adjustment factors for controlling the equilibrium point between acceleration and convergence for ILU(k) and ILUT, respectively. All these ILU preconditioners are characterized and compared in this work, which shows a clear picture and numerical insights for practitioners in the ILU family. The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz equation including very-high-frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three-dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete @math 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 @math 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.