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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 butterfly algorithm for synthetic aperture radar imaging
, 2010
"... Abstract. In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from N data points in provable nearlinear time complexity. This paper seeks to close this gap by proposing an algorith ..."
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Cited by 11 (3 self)
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Abstract. In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from N data points in provable nearlinear time complexity. This paper seeks to close this gap by proposing an algorithm which runs in complexity O(N log N log(1/ɛ)) without making the farfield approximation or imposing the beampattern approximation required by timedomain backprojection, with ɛ the desired pixelwise accuracy. It is based on the butterfly scheme, which unlike the FFT works for vastly more general oscillatory integrals than the discrete Fourier transform. A complete error analysis is provided: the rigorous complexity bound has additional powers of log N and log(1/ɛ) that are not observed in practice. Acknowledgment. LD would like to thank Stefan Kunis for early discussions on error propagation analysis
Fast direct solvers for elliptic partial differential equations
, 2011
"... The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace’s and Helmholtz ’ equations, the equations of elasticity, and timeharmonic Maxwell’s equation. In many areas of science and engineering, ..."
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Cited by 10 (4 self)
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The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace’s and Helmholtz ’ equations, the equations of elasticity, and timeharmonic Maxwell’s equation. In many areas of science and engineering, the cost of solving such problems determines what can and cannot be modeled computationally. Elliptic boundary value problems may be solved either via discretization of the PDE (e.g., finite element methods) or by first reformulating the equation as an integral equation, and then discretizing the integral equation. In either case, one is left with the task of solving a system of
AN O(N) DIRECT SOLVER FOR INTEGRAL EQUATIONS ON THE PLANE
"... Abstract. An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that offdiagonal blocks of certain dense matrices have numerically low ..."
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Cited by 7 (0 self)
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Abstract. An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that offdiagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on “recursive skeletonization ” and “Hierarchically SemiSeparable” (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 107 unknowns can be solved to precision 10−10 using a simple Matlab implementation of the algorithm executed on a single core.
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
Onsurface radiation condition for multiple scattering of waves
 Computer Methods in Applied Mechanics and Engineering, in press
, 2014
"... The formulation of the onsurface radiation condition (OSRC) is extended to handle wave scattering problems in the presence of multiple obstacles. The new multipleOSRC simultaneously accounts for the outgoing behavior of the wave fields, as well as, the multiple wave reflections between the obstac ..."
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Cited by 1 (0 self)
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The formulation of the onsurface radiation condition (OSRC) is extended to handle wave scattering problems in the presence of multiple obstacles. The new multipleOSRC simultaneously accounts for the outgoing behavior of the wave fields, as well as, the multiple wave reflections between the obstacles. Like boundary integral equations (BIE), this method leads to a reduction in dimensionality (from volume to surface) of the discretization region. However, as opposed to BIE, the proposed technique leads to boundary integral equations with smooth kernels. Hence, these Fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities. Moreover, under weak scattering conditions, this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration. Inherited from the original OSRC, the proposed multipleOSRC is generally a crude approximate method. If accuracy is not satisfactory, this approach may serve as a good initial guess or as an inexpensive preconditioner for Krylov iterative solutions of BIE.
A Complete Bibliography of Publications in Journal of Computational Chemistry: 1990–1999
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