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## Recursive Sweeping Preconditioner for the 3D Helmholtz Equation (2015)

Citations: | 1 - 1 self |

### Citations

1048 |
Perfectly matched layer for the absorption of electromagnetic waves,”
- Bérenger
- 1994
(Show Context)
Citation Context ...results are presented in Section 5. Finally, the conclusion and some future directions are provided in Section 6. 2 Problem Formulation Following [7], we assume that the perfectly matched layer (PML) =-=[1, 4, 12]-=- is utilized at part of the boundary where the Sommerfeld radiation condition is specified. The sweeping preconditioner in 2 [7] requires that at least one of the six faces of the domain D = (0, 1)3 i... |

306 |
The multifrontal solution of indefinite sparse symmetric linear equations,
- Duff
- 1983
(Show Context)
Citation Context ...estriction on Pm of the solution v from step 2. The discretized system is a quasi-2D system as b is typically a small constant, so the system can be solved efficiently by the nested dissection method =-=[11, 5, 13]-=-. The first b layers, which are in the PML region of the original problem (2), need to be handled with a slight difference. Define u:,:,1:b = [u T :,:,1, . . . , u T :,:,b] T , f:,:,1:b = [f T :,:,1, ... |

266 |
Nested dissection of a regular finite element mesh.
- George
- 1973
(Show Context)
Citation Context ...estriction on Pm of the solution v from step 2. The discretized system is a quasi-2D system as b is typically a small constant, so the system can be solved efficiently by the nested dissection method =-=[11, 5, 13]-=-. The first b layers, which are in the PML region of the original problem (2), need to be handled with a slight difference. Define u:,:,1:b = [u T :,:,1, . . . , u T :,:,b] T , f:,:,1:b = [f T :,:,1, ... |

256 | A 3D perfectly matched medium from modified Maxwell’s equations with streched coordinates,”
- Chew
- 1994
(Show Context)
Citation Context ...results are presented in Section 5. Finally, the conclusion and some future directions are provided in Section 6. 2 Problem Formulation Following [7], we assume that the perfectly matched layer (PML) =-=[1, 4, 12]-=- is utilized at part of the boundary where the Sommerfeld radiation condition is specified. The sweeping preconditioner in 2 [7] requires that at least one of the six faces of the domain D = (0, 1)3 i... |

123 |
The Multifrontal Method for Sparse Matrix Solution: Theory and Practice,”
- Liu
- 1992
(Show Context)
Citation Context ...estriction on Pm of the solution v from step 2. The discretized system is a quasi-2D system as b is typically a small constant, so the system can be solved efficiently by the nested dissection method =-=[11, 5, 13]-=-. The first b layers, which are in the PML region of the original problem (2), need to be handled with a slight difference. Define u:,:,1:b = [u T :,:,1, . . . , u T :,:,b] T , f:,:,1:b = [f T :,:,1, ... |

46 | Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers,” Multiscale Model.
- Engquist, Ying
- 2011
(Show Context)
Citation Context ...itioner, the method uses the absorbing boundary condition (ABC), which is less effective compared to the PML, and hence the iteration count grows much more rapidly. Since the sweeping preconditioners =-=[6, 7]-=- were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to [15, 14, 18, 16, 17, 19, 2, 3, 20].... |

42 | Advances in Iterative Methods and Preconditioners for the Helmholtz Equation,” - Erlangga - 2008 |

38 | Why it is difficult to solve Helmholtz problems with classical iterative methods.
- Ernst, Gander
- 2012
(Show Context)
Citation Context ...and the standard iterative solvers and/or preconditioners are no longer efficient for such problems. These together make the problem challenging for numerical solution. We refer to the review article =-=[9]-=- by Ernst and Gander for more details on this. Recently in [7], Engquist and Ying developed a sweeping preconditioner using the moving perfectly matched layers (PMLs) and obtained essentially linear s... |

15 |
AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization,”
- Gander
- 2001
(Show Context)
Citation Context ...9] by Ernst and Gander for a rather complete discussion. The discussion here only touches on the methods that share similarity with the sweeping preconditioners. The analytic ILU factorization (AILU) =-=[10]-=- is the first to use incomplete LDU factorizations for preconditioning the Helmholtz equation. Compared to the moving PML sweeping preconditioner, the method uses the absorbing boundary condition (ABC... |

12 | A rapidly converging domain decomposition method for the Helmholtz equation. ArXiv e-prints
- Stolk
- 2012
(Show Context)
Citation Context ...the sweeping preconditioners [6, 7] were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to =-=[15, 14, 18, 16, 17, 19, 2, 3, 20]-=-. In [15], Stolk proposed a domain decomposition algorithm that utilizes suitable transmission conditions based on the PMLs between the subdomains to achieve a near-linear cost. In [14], Poulson et al... |

10 |
Notes on Perfectly Matched Layers (PMLs
- Johnson
- 2010
(Show Context)
Citation Context ...results are presented in Section 5. Finally, the conclusion and some future directions are provided in Section 6. 2 Problem Formulation Following [7], we assume that the perfectly matched layer (PML) =-=[1, 4, 12]-=- is utilized at part of the boundary where the Sommerfeld radiation condition is specified. The sweeping preconditioner in 2 [7] requires that at least one of the six faces of the domain D = (0, 1)3 i... |

6 |
A sweeping preconditioner for timeharmonic Maxwell’s equations with finite elements
- Tsuji, Engquist, et al.
(Show Context)
Citation Context ...the sweeping preconditioners [6, 7] were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to =-=[15, 14, 18, 16, 17, 19, 2, 3, 20]-=-. In [15], Stolk proposed a domain decomposition algorithm that utilizes suitable transmission conditions based on the PMLs between the subdomains to achieve a near-linear cost. In [14], Poulson et al... |

6 | Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem
- Vion, Geuzaine
(Show Context)
Citation Context ...the sweeping preconditioners [6, 7] were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to =-=[15, 14, 18, 16, 17, 19, 2, 3, 20]-=-. In [15], Stolk proposed a domain decomposition algorithm that utilizes suitable transmission conditions based on the PMLs between the subdomains to achieve a near-linear cost. In [14], Poulson et al... |

4 | A source transfer domain decomposition method for Helmholtz equations in unbounded domain part II: Extensions
- Chen, Xiang
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4 | A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations
- Poulson, Engquist, et al.
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4 | Sweeping preconditioners for elastic wave propagation with spectral element methods
- Tsuji, Poulson, et al.
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4 |
A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s Equations
- Tsuji, Ying
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3 | The method of polarized traces for the 2d helmholtz equation, submitted
- Zepeda-Nunez, Demanet
- 2014
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