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Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 46 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency 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 threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
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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Cited by 4 (3 self)
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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.
The method of polarized traces for the 2D Helmholtz equation. ArXiv eprints
, 2014
"... We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N ..."
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Cited by 3 (0 self)
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We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define “polarized traces”, i.e., up and downgoing waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green’s functions in an embarrassingly parallel way (the offline part), and incomplete Green’s formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive lowrank partitioning of the integral kernels is used to speed up their application to interface data. The method uses secondorder finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of offdiagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. 1
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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Cited by 1 (1 self)
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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.
Simulation and Design Optimization for Linear Wave Phenomena on Metamaterials
, 2011
"... A ccepted by.......................... Nic a jiconstantinou Associate Professor of ec anical Engineering ..."
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A ccepted by.......................... Nic a jiconstantinou Associate Professor of ec anical Engineering
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"... Krylov subspace methods for time modeling of photonic crystals Bachelor assignment ..."
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Krylov subspace methods for time modeling of photonic crystals Bachelor assignment
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.
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"... A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional Giulio Ciraolo∗ † Francesco Gargano ∗ Vincenzo Sciacca∗ ..."
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A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional Giulio Ciraolo∗ † Francesco Gargano ∗ Vincenzo Sciacca∗
Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1081286515569266
"... mms.sagepub.com Compact secondorder timedomain perfectly matched layer formulation for elastic wave propagation in two dimensions ..."
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mms.sagepub.com Compact secondorder timedomain perfectly matched layer formulation for elastic wave propagation in two dimensions