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SWEEPING PRECONDITIONERS FOR ELASTIC WAVE PROPAGATION WITH SPECTRAL ELEMENT METHODS
, 2013
"... Abstract.We present a parallel preconditioning method for the iterative solution of the timeharmonic elastic wave equation which makes use of higherorder spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently appr ..."
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Abstract.We present a parallel preconditioning method for the iterative solution of the timeharmonic elastic wave equation which makes use of higherorder spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDLT factorization. Both sequential and parallel versions of the algorithm are discussed and results for largescale problems from exploration geophysics are presented.
A PARALLEL SWEEPING PRECONDITIONER FOR HETEROGENEOUS 3D HELMHOLTZ EQUATIONS∗
"... Abstract. A parallelization of a sweeping preconditioner for 3D Helmholtz equations without internal resonance is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ2N4/3) and O(γN logN), where γ(ω) ..."
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Abstract. A parallelization of a sweeping preconditioner for 3D Helmholtz equations without internal resonance is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ2N4/3) and O(γN logN), where γ(ω) denotes the modestly frequencydependent number of grid points per Perfectly Matched Layer. Several computational and memory improvements are introduced relative to using blackbox sparsedirect solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for highfrequency problems distributed over thousands of cores. Two opensource packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.
Recursive Sweeping Preconditioner for the 3D Helmholtz Equation
, 2015
"... This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively t ..."
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This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the nonrecursive 3D sweeping preconditioner, the setup cost of this new approach drops from O(N4/3) to O(N), the application cost per iteration drops from O(N logN) to O(N), and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach.
An Effective Preconditioner for a PML System For Electromagnetic Scattering Problem
"... In this work we are concerned with an efficient numerical solution of a perfectly matched layer (PML) system for a Maxwell scattering problem. The PML system is discretized by the edge nite element method, resulting in a symmetric but indenite complex algebraic system. When the real and imaginary p ..."
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In this work we are concerned with an efficient numerical solution of a perfectly matched layer (PML) system for a Maxwell scattering problem. The PML system is discretized by the edge nite element method, resulting in a symmetric but indenite complex algebraic system. When the real and imaginary parts are considered independently, the complex algebraic system can be further transformed into a real generalized saddlepoint system with some special structure. Based on an crucial observation to its Schur complement, we construct a symmetric and positive denite block diagonal preconditioner for the saddlepoint system. Numerical experiments are presented to demonstrate the effectiveness and robustness of the new preconditioner.
Additive Sweeping Preconditioner for the Helmholtz Equation
, 2015
"... We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the in ..."
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We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the intermediate waves. This approach can be viewed as an effective approximation of an additive decomposition of the solution operator. When combined with the standard GMRES solver, the iteration number is essentially independent of the frequency. Several numerical examples are tested to show the efficiency of this new approach.