K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole accelerated preconditioned iterative methods for three-dimensional potential integral equations of the first kind," SIAM J. Sci. Stat. Comput., vol. AP-15, pp. 713--735, May 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... methods, and matrix sparsification techniques, like fast multipole and multilevel methods, have been used to create very fast boundary element codes [4, 8, 2] Fast multipole based codes for solving potential problems with i kernels are now commonly used in a variety of engineering applications [3]. What is now of primary research interest is developing sparsification procedures for boundary element matrices which are capable of solving potential problems with relatively general kernels, at least including i and ik, for a wide range of kr [13, 14, 11 12, 8, 15] Such a direction parallels ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole Acceler- ated Preconditioned Iterative Methods for Three-Dimensional Potential Integral Equations of the First Kind," SIAM J. on Sci. and Star. Comp., May 1994, Vol. 15, No. 3, p713-735.

.... multipole method for the rapid calculation of the matrix vector products; see, for example, Rokhlin [20] Greenbaum, Greengard Mayo [8] A comparison of CGN, GMRES and CGS methods can be found in [16] Recently some preconditioners have been proposed (see Vavasis [22] Nabors, Korsmeyer White [15]) for the linear system (1.1) arising from three dimensional problems. 1 2 Y. Yan Another typical dense linear system is from singular boundary integral equations with logarithmic, Cauchy or hypersingular kernels and has the form (see, for example, 19] 4] 7] C B)w = b (1.2) where C is ....

....system (1.2) have also been studied (see Reichel [17] Hebeker [11] Graham Yan [7] but have received much less attention. In the iterative methods of this paper, the most difficult task is to find appropriate preconditioners. These preconditioners are quite different from those proposed in [15] and [22] and are constructed by using information only from the linear system (1.1) or (1.2) itself. This idea is different from those in the two grid and multigrid methods, where preconditioners are built by employing information of a linear system at a low level. When the preconditioners are ....

K. Nabors, T. Korsmeyer and J. White, Multipole-accelerated preconditioned iterative methods for solving three-dimensional mixed first and second kind integral equations, Proc. 1992 Copper Mountain Conference on Iterative Methods.

.... and matrix sparsification techniques, like fast multipole and multilevel methods, have been used to create very fast boundary element codes [4, 8, 2] Fast multipole based codes for solving potential problems with 1 r kernels are now commonly used in a variety of engineering applications [3]. What is now of primary research interest is developing sparsification procedures for boundary element matrices which are capable of solving potential problems with relatively general kernels, at least including 1 r and e ikr r for a wide range of kr [13, 14, 11, 12, 8, 15] Such a ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole Accelerated Preconditioned Iterative Methods for Three-Dimensional Potential Integral Equations of the First Kind," SIAM J. on Sci. and Stat. Comp., May 1994, Vol. 15, No. 3, p713-735.

.... Theta(n log n) time. Parallel formulations of hierarchical methods have been explored in the context of particle dynamics for regular distributions [1, 12, 6, 8, 5] and irregular distributions [3, 4, 2, 9, 11, 10] These techniques have been explored to solve integral equations by Nabors et al. [7]. In [3] we presented a highly scalable parallel formulation of a dense matrix vector product for unstructured and adaptive discretizations based on hierarchical methods. Here, we present the performance and accuracy of a GMRES solver based on this parallel mat vec. We also present techniques for ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White. Multipole accelerated preconditioned iterative methods for three-dimensional potential integral equations of the first kind. J. on Sci. and Stat. Comp., 15(3):713--735, May 1994.

....constructed, preconditioners must be derived from the hierarchical domain representation. Furthermore, the preconditioning strategies must be highly parallelizable. Since the early work of Rokhlin[16] relatively little work has been done on dense hierarchical solvers even in the serial context [14, 17, 22, 3]. In this paper, we investigate the accuracy and convergence of a GMRES solver built around a parallel hierarchical matrix vector product. We investigate the impact of various parameters on accuracy and performance. We propose two preconditioning strategies for accelerating the convergence of the ....

....The treecode developed here is highly modular in nature and provides a general framework for solving a variety of dense linear systems. Even in the serial context, relatively little work has been done since the initial work of Rokhlin[16] Other prominent pieces of work were in this area include [14, 17, 22, 3]. To the best of our knowledge, the treecode presented in this paper is among the first parallel multilevel solver preconditioner toolkit. We are currently extending the hierarchical solver to scattering problems in electromagnetics [17, 16, 22, 21, 3] The free space Green s function for the ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White. Multipole accelerated preconditioned iterative methods for three-dimensional potential integral equations of the first kind. J. on Sci. and Stat. Comp., 15(3):713--735, May 1994.

....less efficient, as the grid representation is introduced throughout space, even where no panels are present. Thus, whereas for a problem containing panels, the fast multipole algorithm can perform a potential evaluation for all of the panels in operations, regardless of the panel distribution [32], no such guarantee (a) b) c) d) Fig. 9. Several realistic capacitance extraction problems. a) The woven bus example (woven 5 2 5) b) The comb drive example (comb) c) The via example (via) d) The SRAM example (SRAM) is available for the precorrected FFT algorithm. However, it is ....

....but more distant interactions are approximated by extrapolation, convolution, and then interpolation using the grid. To demonstrate that the errors due to using the grid are well controlled, we present an empirical error study based on an analytically solvable potential problem borrowed from [32]. If (1) is solved on a sphere with given potential (18) the analytically computable charge distribution is (19) To estimate the error introduced by the grid approximations in the precorrected FFT method, the sphere can be discretized, as in Fig. 6, and the charges on each panel computed. The ....

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K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole accelerated preconditioned iterative methods for three-dimensional potential integral equations of the first kind," SIAM J. Sci. Stat. Comput., vol. AP-15, pp. 713--735, May 1994.

....2700n. So, although the fast multipole algorithm is asymptotically faster, it is only of practical significance for extremely large n. It should be noted that the above result is NOT general. The fast multipole algorithm retains its linear time behavior even in the arbitrarily inhomogenous case [5]. The precorrected FFT method is only suitable when the distribution can be made to look homogenous. More detailed experiments are required to better understand these pragmatic issues. ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole Accelerated Preconditioned Iterative Methods for Three-Dimensional Potential Integral Equations of the First Kind," To Appear, SIAM J. on Sci. and Star. Comp.

....2700n. So, although the fast multipole algorithm is asymptotically faster, it is only of practical significance for extremely large n. It should be noted that the above result is NOT general. The fast multipole algorithm retains its linear time behavior even in the arbitrarily inhomogenous case [5]. The precorrected FFT method is only suitable when the distribution can be made to look homogenous. More detailed experiments are required to better understand these pragmatic issues. ....

K. Nabors, F. T. Korsmeyer, F. T. Leighton, and J. White, "Multipole Accelerated Preconditioned Iterative Methods for Three-Dimensional Potential Integral Equations of the First Kind," To Appear, SIAM J. on Sci. and Stat. Comp.

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