To reduce computational complexity and memory requirement for 3-D elastodynamics using the boundary element method (BEM), a multi-level fast multipole BEM (FM-BEM) is proposed. The diagonal form for the expansion of the elastodynamic fundamental solution is used, with a truncation parameter adjusted to the subdivision level, a feature necessary for achieving optimal computational efficiency. Both the single-level and multi-level forms of the elastodynamic FM-BEM are considered, with emphasis on the latter. Crucial implementation issues, including the truncation of the multipole expansion, the optimal number of levels, the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments. A complexity analysis for both the single-level and multi-level versions is conducted. The correctness and computational performances of the proposed elastodynamic FMM are demonstrated on numerical examples, featuring up to O(10 6 ) DOFs run on a single-processor PC and including the diffraction of an incident P plane wave by a semi-spherical or semi-ellipsoidal canyon, representative of topographic site effects.Keywords: Fast multipole method; Boundary element method; 3-D elastodynamics; Topographic site effects IntroductionThe boundary element method (BEM), pioneered in the sixties [6,41], is a mesh reduction method, subject to restrictive constitutive assumptions but yielding highly accurate solutions. It is in particular well suited to deal with unbounded-domain idealizations commonly used in e.g. acoustics [50], electromagnetics [35,39] or seismology [7,22]. In contrast with domain discretization methods, artificial boundary conditions [18] are not needed for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [24,51].However, in traditional boundary element (BE) implementations, the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix, with set-up and solution times rapidly increasing with the problem size N . It is thus essential to develop alternative, faster strategies that allow to still exploit the known advantages of BEMs when large N prohibit the use of traditional implementations. Fast BEMs, i.e. BEMs of complexity lower than that of traditional BEMs, appeared around 1985 with an iterative integral-equation [19,20], in the context of many-particle simulations. The FMM then naturally led to fast multipole boundary element methods (FM-BEMs), whose scope and capabilities have rapidly progressed, especially in connection with application in electromagnetics [21,31,32,53], but also in other fields including acoustics [14,36,48] and computational mechanics [30]. Many of these investigations are summarized in a review article by Nishimura [37]. The FMM, as well as other fast BEM approaches [23,27,55,56], intrinsically relies upon an iterative solution approach for the linear system of discretized BEM equations, with solution times typically of order O(N lo...
S U M M A R YThe analysis of seismic wave propagation and amplification in complex geological structures raises the need for efficient and accurate numerical methods. The solution of the elastodynamic equations using traditional boundary element methods (BEMs) is greatly hindered by the fully-populated nature of the matrix equations arising from the discretization. In a previous study limited to homogeneous media, the present authors have established that the fast multipole method (FMM) reduces the complexity of a 3-D elastodynamic BEM to N log N per GMRES iteration and demonstrated its effectiveness on 3-D canyon configurations. In this paper, the frequency-domain FM-BEM methodology is extented to 3-D elastic wave propagation in piecewise homogeneous domains in the form of a FM-accelerated multi-region BE-BE coupling approach. This new method considerably enhances the capability of the BEM for studying the propagation of seismic waves in 3-D alluvial basins of arbitrary geometry embedded in semi-infinite media. Several fully 3-D examples (oblique SV -waves) representative of such configurations validate and demonstrate the capabilities of the multi-domain FM approach. They include comparisons with available (low-frequency) results for various types of incident wavefields and time-domain results obtained by means of Fourier synthesis.
We propose an algorithm to compute an approximate singular value decomposition (SVD) of leastsquares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency timeharmonic wave equation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, subsurface imaging, impedance tomography, non-destructive evaluation, and diffuse optical tomography). We consider small perturbations of the background medium and, by invoking the Born approximation, we obtain a linear least-squares problem. The scheme we describe in this paper constructs an approximate SVD of the Born operator (the operator in the linearized least-squares problem). The main feature of the method is that it can accelerate the application of the Born operator to a vector. If N ω is the number of illumination frequencies, N s the number of illumination locations, N d the number of detectors, and N the discretization size of the medium perturbation, a dense singular value decomposition of the Born operator requires O(min(N s N ω N d , N)] 2 × max(N s N ω N d , N)) operations. The application of the Born operator to a vector requires O(N ω N s µ(N)) work, where µ(N) is the cost of solving a forward scattering problem. We propose an approximate SVD method that, under certain conditions, reduces these work estimates significantly. For example, the asymptotic cost of factorizing and applying the Born operator becomes O(µ(N)N ω). We provide numerical results that demonstrate the scalability of the method.
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