Papers by Leslie Greengard
We present a fast direct solver for boundary integral equations on complex surfaces in three dime... more We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions, using an extension of the recently introduced strong recursive skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $O(N)$ time, where $N$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted, adaptive octree data structure and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate application to a variety of problems in computational physics. We conce...
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We have developed a fast direct solver for structured linear systems based on multilevel matrix c... more We have developed a fast direct solver for structured linear systems based on multilevel matrix compression. Starting with a hierarchically block-separable matrix [2], we embed an approximation of the original matrix into a larger, but highly structured sparse one. The resulting representation allows for efficient storage, fast matrix-vector multiplication, fast matrix factorization, and fast application of the inverse. The algorithm proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green’s functions for the Laplace or lowfrequency Helmholtz equations, both phases typically have complexity O(N) in two dimensions, where N is the number of discretization points. In our current three-dimensional implementation, the corresponding costs are O(N3/2) and O(N logN) for precomputation and solution, respec...
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Communications on Pure and Applied Mathematics, 2020
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SIAM Journal on Scientific Computing, 2020
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SIAM Journal on Numerical Analysis, 2016
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Journal of Computational Physics, 2019
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Journal of Computational Physics, 2019
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We present a fast direct solver for boundary integral equations on complex surfaces in three dime... more We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions, using an extension of the recently introduced strong recursive skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $O(N)$ time, where $N$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted, adaptive octree data structure and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate application to a variety of problems in computational physics. We conce...
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IEEE Transactions on Antennas and Propagation, 2019
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SIAM Journal on Scientific Computing, 2020
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Journal of Mathematical Physics, 2017
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SIAM Journal on Numerical Analysis, 2016
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Applied and Computational Harmonic Analysis, 2016
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Journal of Computational Physics, 2019
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Journal of Computational Physics, 2017
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IEEE Transactions on Antennas and Propagation, 2019
Bookmarks Related papers MentionsView impact
SIAM Journal on Scientific Computing, 2016
Bookmarks Related papers MentionsView impact
Applied and Computational Harmonic Analysis, 2016
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Journal of Computational Physics, 2016
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Journal of Computational Physics, 2017
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Papers by Leslie Greengard