Fast algorithms for Helmholtz Green's functions

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 464, Issue 2100, 2008, Pages 3301-3326

  Beylkin, Gregory ^a   Kurcz, Christopher ^a   Monzón, Lucas ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Dirichlet and Neumann boundary conditions; Fast adaptive solvers; Lattice sums; Quasi periodic Green's functions; Separated representations; Unequally spaced fast Fourier transform

Indexed keywords

ADAPTIVE ALGORITHMS; ALGORITHMS; APPROXIMATION ALGORITHMS; BOOLEAN FUNCTIONS; BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; FAST FOURIER TRANSFORMS; FOURIER TRANSFORMS; HELMHOLTZ EQUATION; MATHEMATICAL OPERATORS; PROBABILITY DENSITY FUNCTION; THEOREM PROVING;

DIRICHLET AND NEUMANN BOUNDARY CONDITIONS; FAST ADAPTIVE SOLVERS; LATTICE SUMS; QUASI-PERIODIC GREEN'S FUNCTIONS; SEPARATED REPRESENTATIONS; UNEQUALLY SPACED FAST FOURIER TRANSFORM;

GREEN'S FUNCTION;

EID: 54849442246     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2008.0161     Document Type: Article

References (40)

1. Abramowitz, M. & Stegun, I. A. 1970 Handbook of mathematical functions, 9th edn. New York, NY: Dover Publications.
2. Berry, M. 1981 Quantizing a classically ergodic system: Sinai's billiard and the KKR method. Ann. Phys. 131, 163-216. (doi: 10.1016/0003-4916(81)90189-5)
3. Beylkin, G. 1995 On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal. 2, 363-381. (doi:10.1006/acha.1995.1026)
4. Beylkin, G. & Mohlenkamp, M. J. 2002 Numerical operator calculus in higher dimensions. Proc. Natl Acad. Sci. USA 99, 10246-10251. (doi:10.1073/pnas.112329799)
5. Beylkin, G. & Mohlenkamp, M. J. 2005 Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26, 2133-2159. (doi:10.1137/040604959)
6. Beylkin, G. & Monzón, L. 2005 On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19, 17 48. (doi:10.1016/j.acha.2005.01.003)
7. Beylkin, G., Cramer, R., Fann, G. & Harrison, R. 2007 Multiresolution separated representations of singular and weakly singular operators. Appl. Comput. Harmon. Anal. 23, 235-253. (doi:10.1016/j.acha.2007.01.001)
8. Beylkin, G., Cheruvu, V. & Pérez, F. 2008 Fast adaptive algorithms in the non-standard form for multidimensional problems. Appl. Comput. Harmon. Anal. 24, 354-377. (doi:10.1016/j.acha.2007.08.001)
9. Beylkin, G., Kurcz, C. & Monzón, L. Submitted. Fast convolution with the free space Helmholtz Green's function.
10. Brillouin, L. 1953 Wave propagation in periodic structures. New York, NY: Dover.
11. Catti, M. 1978 Electrostatic lattice energy in ionic crystals: optimization of the convergence of Ewald series. Acta Crystallogr. A 34, 974-979. (doi:10.1107/S0567739478001990)
12. Cruickshank, D., Juretschke, H. & Kato, N. (eds) 1992 P.P. Ewald and his dynamical theory of X-ray diffraction. Oxford, UK: Oxford University Press.
13. Dienstfrey, A., Hang, F. & Huang, J. 2001 Lattice sums and the two-dimensional, periodic Green's function for the Helmholtz equation. Proc. R. Soc. A 457, 67-85. (doi:10.1098/rspa.2000.0656)
14. Dutt, A. & Rokhlin, V. 1993 Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14, 1368 1393. (doi:10.1137/0914081)
15. Ewald, P. 1913 Contributions to the theory of the interferences of X-rays in crystals. Phys. Z 14, 465-472.
16. Ewald, P. 1921 Die berechnung optischer und elektrostatischer gitterpotentiale. Ann. Phys. 64, 253 287. (doi:10.1002/andp. 19213690304)
17. Gel'fand, I. M. & Shilov, G. E. 1964 Generalized functions, vol. 1. Properties and operations (transl. from the Russian by Eugene Salctan). New York, NY: Academic Press.
18. Classer, M. & Zucker, I. 1980 Lattice sums. In Theoretical chemistry: advances and perspectives (eds H. Eyring & D. Heriderson), pp. 67-139. New York, NY: Academic Press.
19. Grafakos, L. 2004 Classical and modern Fourier analysis. Upper Saddle River, NJ: Pearson Education, Inc.
20. Greengard, L. & Strain, J. 1991 The fast Gauss transform. SIAM J. Sci. Stat. Comput. 12, 79-94. (doi:10.1137/0912004)
21. Greengard, L. & Sun, X. 1998 A new version of the fast Gauss transform. In Proc. Int. Congress of Mathematicians, Berlin, Germany, 18-27 August, vol. III (Extra vol.), pp. 575-584. Bielefeld, Germany: University of Bielefeld.
22. Ham, F. & Segall, B. 1961 Energy bands in periodic lattice - Green's function method. Phys. Rev. 124, 1786 1796. (doi:10.1103/PhysRev. 124.1786)
23. Harrison, R., Farm, G., Yanai, T. & Beylkin, G. 2003 Multiresolution quantum chemistry in multiwavelet bases. In Computational science-ICCS 2003, vol. 2660 (eds P. M. A. Sloot, D. Abramson, A. V. Bogdanov, J. Dongarra, A. Y. Zomaya & Y. E. Gorbachev), pp. 103-110. Lecture Notes in Computer Science. Berlin, Germany: Springer.
24. Harrison, R., Fann, G., Yanai, T., Gan, Z. & Beylkin, G. 2004 Multiresolution quantum chemistry: basic theory and initial applications. J. Chem. Phys. 121, 11 587-11 598. (doi:10.1063/1.1791051)
25. Joannopoulos, J., Meade, R. & Winn, J. 1995 Photonic crystals: modeling the flow of light. Princeton, NJ: Princeton University Press.
26. Jordan, K. E., Richter, G. R. & Sheng, P. 1986 An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures. J. Comput. Phys. 63, 222-235. (doi:10.1016/0021- 9991(86)90093-8)
27. Keller, J. B. 1953 The scope of the image method. Comm. Pure Appl. Math. 6, 505-512. (doi:10.1002/cpa.3160060406)
28. Kittel, C. 1986 Introduction to solid state physics, 6th edn. New York, NY: Wiley.
29. Kohn, W. & Rostoker, N. 1954 Solution of the Schröndinger equation in periodic lattices with an application to metallic lithium. Phys. Rev. 94, 1111-1120. (doi:10.1103/PhysRev.94.1111)
30. Lee, J.-Y. & Greengard, L. 2005 The type 3 nonuniform FFT and its applications. J. Comput. Phys. 206, 1-5. (doi:10.1016/j.jcp.2004. 12.004)
31. Linton, C. M. 1998 The Green's function for the two-dimensional Helmholtz equation in periodic domains. J. Eng. Math. 33, 377-402. (doi:10.1023/A:1004377501747)
32. McPhedran, R. C., Nicorovici, N. A., Botten, L. C. & Grubits, K. A. 2000 Lattice sums for gratings and arrays. J. Math. Phys. 41, 7808-7816. (doi:10.1063/1.1310361)
33. McPhedran, R. C., Nicorovici, N. A. & Botten, L. C. 2005 Neumann series and lattice sums. J. Math. Phys. 46, 083509. (doi:10.1063/1.1998827)
34. Moroz, A. 2006 Quasi-periodic Green's functions of the helmholtz and laplace equations. J. Phys. A Math. Gen. 39, 11 247-11 282. (doi:10.1088/0305-4470/39/36/009)
35. Oroskar, S., Jackson, D. & Wilton, D. 2006 Efficient computation of the 2D periodic Green's function using the Ewald method. J. Comput. Phys. 219, 899-911. (doi:10.1016/j.jcp.2006.06.050)
36. Soukoulis, C. (ed.) 1992 Photonic band gaps and localization. New York, NY: Plenum Press.
37. Strain, J. 1991 The fast Gauss transform with variable scales. SIAM J. Sci. Statist. Comput. 12, 1131-1139. (doi:10.1137/0912059)
38. Yanai, T., Fann, G., Gan, Z., Harrison, R. & Beylkin, G. 2004a Multiresolution quantum chemistry: analytic derivatives for Hartree - Fock and density functional theory. J. Chem. Phys. 121, 2866-2876. (doi:10.1063/1.1768161)
39. Yanai, T., Fann, G., Gan, Z., Harrison, R. & Beylkin, G. 20046 Multiresolution quantum chemistry: Hartree-Fock exchange. J. Chem. Phys. 121, 6680-6688. (doi:10.1063/1.1790931)
40. Yasurnoto, K. & Yoshitomi, K. 1999 Efficient calculation of lattice sums for free-space periodic Green's function. IEEE Trans. Antennas Propagation 47, 1050 1055. (doi:10.1109/8.777130)