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**11 - 19**of**19**### unknown title

"... 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 near-linear time complexity. This paper seeks to close this gap by proposing an algorith ..."

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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 near-linear time complexity. This paper seeks to close this gap by proposing an algorithm which runs in complexity O(N logN log(1/)) without making the far-field approximation or imposing the beam pattern approximation required by time-domain backprojection, with 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 logN and log(1/) that are not observed in practice.

### doi: 10.1017/S0962492906410011 Printed in the United Kingdom

"... Fast direct solvers for integral equations in complex three-dimensional domains ..."

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Fast direct solvers for integral equations in complex three-dimensional domains

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"... Abstract. The butterfly algorithm is a fast algorithm which approximately evaluates a dis-crete analogue of the integral transform Rd K(x, y)g(y)dy at large numbers of target points when the kernel, K(x, y), is approximately low-rank when restricted to subdomains satisfying a certain simple geometri ..."

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Abstract. The butterfly algorithm is a fast algorithm which approximately evaluates a dis-crete analogue of the integral transform Rd K(x, y)g(y)dy at large numbers of target points when the kernel, K(x, y), is approximately low-rank when restricted to subdomains satisfying a certain simple geometric condition. In d dimensions with O(Nd) quasi-uniformly distributed source and target points, when each appropriate submatrix of K is approximately rank-r, the running time of the algorithm is at most O(r2Nd logN). A parallelization of the butterfly algorithm is intro-duced which, assuming a message latency of α and per-process inverse bandwidth of β, executes in at most O(r2N d

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"... Abstract. The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems from bounded scatterers. This paper presents the directional precondi-tioner for the linear systems of the boundary integral method in two dimensions. This new precondi-tioner build ..."

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Abstract. The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems from bounded scatterers. This paper presents the directional precondi-tioner for the linear systems of the boundary integral method in two dimensions. This new precondi-tioner builds a data-sparse approximation of the integral operator, transforms it into a sparse linear system, and computes an approximate inverse with efficient sparse linear algebra algorithms. This preconditioner is efficient and results in small and almost frequency-independent iteration counts for nonresonant scatterers when combined with standard iterative solvers. Numerical results are provided to demonstrate the effectiveness of the new preconditioner.

### scattering from obstacles in

"... boundary integral equation solution of high frequency wave ..."

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### Shifted Laplacian based multigrid preconditioners for solving indefinite Helmholtz equations

"... ar ..."

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### Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics∗

"... This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic prob-lems in semi-infinite media. The main concepts u ..."

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This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic prob-lems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form ker-nel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evalua-tion of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s ten-sor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process. 1

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"... Abstract. We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sp ..."

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Abstract. We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson–Rokhlin method developed for 2D boundary integral equations and 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 functions for the Laplace or low-frequency Helmholtz equa-tions, both phases typically have complexity O(N) in two dimensions, where N is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are O(N3/2) and O(N logN) for precomputation and solution, respectively. Extensive numerical exper-iments show a speedup of ∼100 for the solution phase over modern fast multipole methods; however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with ill-conditioned linear systems or when the same system is to be solved with multiple right-hand sides. Our algorithm is implemented in Fortran and freely available.