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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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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
BUTTERFLY FACTORIZATION∗
"... Abstract. The paper introduces the butterfly factorization as a datasparse approximation for the matrices that satisfy a complementary lowrank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries o ..."
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Abstract. The paper introduces the butterfly factorization as a datasparse approximation for the matrices that satisfy a complementary lowrank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N × N matrix, the resulting factorization is a product of O(logN) sparse matrices, each with O(N) nonzero entries. Hence, it can be applied rapidly in O(N logN) operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.
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"... Abstract. The paper introduces the butterfly factorization as a datasparse approximation for the matrices that satisfy a complementary lowrank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries o ..."
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Abstract. The paper introduces the butterfly factorization as a datasparse approximation for the matrices that satisfy a complementary lowrank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N × N matrix, the resulting factorization is a product of O(logN) sparse matrices, each with O(N) nonzero entries. Hence, it can be applied rapidly in O(N logN) operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.
unknown title
"... Abstract. This paper introduces a parallel directional fast multipole method (FMM) for solving Nbody problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive lowrank criterion than that of the ..."
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Abstract. This paper introduces a parallel directional fast multipole method (FMM) for solving Nbody problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive lowrank criterion than that of the lowfrequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between p processes, the overall runtime is essentially ON logN/p + p. By the structure of the lowrank criterion, we are able to avoid communication at the top of the octree. We demonstrate the effectiveness of our parallelization on several challenging models.
A PARALLEL DIRECTIONAL FAST MULTIPOLE METHOD∗
"... Abstract. This paper introduces a parallel directional fast multipole method (FMM) for solving Nbody problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive lowrank criterion than that of the ..."
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Abstract. This paper introduces a parallel directional fast multipole method (FMM) for solving Nbody problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive lowrank criterion than that of the lowfrequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We propose a simple partition at a fixed level of the octree and show that, if the partitions are properly balanced between p processes, the overall runtime is essentially ON logN/p + p. By the structure of the lowrank criterion, we are able to avoid communication at the top of the octree. We demonstrate the effectiveness of our parallelization on several challenging models.