This article reviews several fast algorithms for boundary integral equations. After a brief introduction of the boundary integral equations for the Laplace and Helmholtz equations, we discuss in order the fast multipole method and its kernel independent variant, the hierarchical matrix framework, the wavelet based method, the high frequency fast multipole method, and the recently proposed multidirectional algorithm.
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References
C. R. Anderson. An implementation of the fast multipole method without multipoles. SIAM J. Sci. Statist. Comput., 13(4):923–947, 1992.
A. Averbuch, E. Braverman, R. Coifman, M. Israeli, and A. Sidi. Efficient computation of oscillatory integrals via adaptive multiscale local Fourier bases. Appl. Comput. Harmon. Anal., 9(1):19–53, 2000.
G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and numerical algorithms. I. Comm. Pure Appl. Math., 44(2):141–183, 1991.
E. Bleszynski, M. Bleszynski, and T. Jaroszewicz. AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems. Radio Science, 31:1225–1252, 1996.
N. Bojarski. K-space formulation of the electromagnetic scattering problems. Technical report, Air Force Avionic Lab. Technical Report AFAL-TR-71-75, 1971.
S. Börm, L. Grasedyck, and W. Hackbusch. Hierarchical matrices. Technical Report 21, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2003.
O. P. Bruno and L. A. Kunyansky. A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications. J. Comput. Phys., 169(1):80–110, 2001.
H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao. A wideband fast multipole method for the Helmholtz equation in three dimensions. J. Comput. Phys., 216(1):300–325, 2006.
H. Cheng, L. Greengard, and V. Rokhlin. A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys., 155(2):468–498, 1999.
W. Chew, E. Michielssen, J. M. Song, and J. M. Jin, editors. Fast and efficient algorithms in computational electromagnetics. Artech House, Inc., Norwood, MA, USA, 2001.
D. Colton and R. Kress. Inverse acoustic and electromagnetic scattering theory, volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, second edition, 1998.
I. Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
B. Engquist and L. Ying. Fast directional multilevel algorithms for oscillatory kernels. SIAM J. Sci. Comput., 29(4):1710–1737 (electronic), 2007.
S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin. A theory of pseudoskeleton approximations. Linear Algebra Appl., 261:1–21, 1997.
L. Greengard. The rapid evaluation of potential fields in particle systems. ACM Distinguished Dissertations. MIT Press, Cambridge, MA, 1988.
L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys., 73(2):325–348, 1987.
L. Greengard and J. Strain. A fast algorithm for the evaluation of heat potentials. Comm. Pure Appl. Math., 43(8):949–963, 1990.
W. Hackbusch. Integral equations, volume 120 of International Series of Numerical Mathematics. Birkhäuser Verlag, Basel, 1995. Theory and numerical treatment, Translated and revised by the author from the 1989 German original.
W. Hackbusch and Z. P. Nowak. On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math., 54(4):463–491, 1989.
R. Kress. Linear integral equations, volume 82 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1999.
S. Mallat. A wavelet tour of signal processing. Academic Press Inc., San Diego, CA, 1998.
P. G. Martinsson and V. Rokhlin. A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys., 205(1):1–23, 2005.
K. Nabors, F. Korsmeyer, F. Leighton, and J. K. White. Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM Journal on Scientific and Statistical Computing, 15:713– 735, 1994.
N. Nishimura. Fast multipole accelerated boundary integral equation methods. Applied Mechanics Reviews, 55(4):299–324, 2002.
V. Rokhlin. Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys., 86(2):414–439, 1990.
V. Rokhlin. Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmon. Anal., 1(1):82–93, 1993.
L. Ying, G. Biros, and D. Zorin. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys., 196(2):591–626, 2004.
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Ying, L. (2009). Fast Algorithms for Boundary Integral Equations. In: Engquist, B., Lötstedt, P., Runborg, O. (eds) Multiscale Modeling and Simulation in Science. Lecture Notes in Computational Science and Engineering, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88857-4_3
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