On the resolution of fredholm integral equations by multigrid domain separation techniques
DOI:
https://doi.org/10.22199/S07160917.1998.0001.00008Keywords:
Nyström methods, Multigrid methorls, Domain Reduction methods, Frequency Decomposition methods, Parallel computingAbstract
We introduce a technique for split discretizations of Fredholm integral equations of the second kind by means of projections on some subsets of quadrature nades. The technique has its roots in the traditional framework of integral operators of the second kind. We combine this approach with two level methods and generate a numerical parallel scheme.
References
[1] R. Aggarwal, D. Dellwo and M. Friedman, Parallel solution of Fredholm integral equations of the second kind by accelerated projection methods, Parallel Computing, Vol. 19. pp. 1105-1115, (1993).
[2] Ph. Anselone. Collectively Compact Operator- Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs. (1971).
[3] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind. Cambridge, University Press (1997).
[4] J. Bramble, Multigrid Methods, Pitman, (1993).
[5] M. Campos and C. Pérez, Paralelización de técnicas para ecuaciones integrales de Fredholm de segunda especie por métodos de cuadratura y multimallas, Departamento de Ingeniería Matemática, Universidad de Concepción, Technical Report 97-09, (1997).
[6] M. Campos and C. Pérez. Tratamiento numérico de ecuaciones integrales mediante técnicas de separación de dominios, Departamento de Ingeniería Matemática, Universidad de Concepción, Technical Report
97-15, (1997).
[7] M. Campos and C. Pérez, On the use of domain separation techniques for solving Fredholm inteqral equations of the second kind, in preparation, ( 1998).
[8] T. Chan and R. Tuminaro, Analysis of a parallel multigrid algorithm., Procceding of the Fourth Copper Mountain conference on multigrid methods, SIAM, pp. 66-86, (1989).
[9] C. Douglas and C. Douglas, A unified convergence theory for abstract multigrid or multilevel methods, serial and parallel, SIAM .J. Num. An., Vol. 30. pp. 136-158, (1993).
[10] C. Douglas and W. Miranker, Constructive interference in parallel algorithms, SIAM J. Num. An., Vol. 25, pp. 376-398, (1988).
[11] I. Goldbcrg, I. Koltracht and P. Lancastcr, Second order parallel algorithms for Fredholm integral equations with continuous displacement kernels, Int. Op. Th., Vol. 10, pp .. 577-594, (1987).
[12] W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, (1985).
[13] W. Hackbusch, The frequency decomposition multi-grid method, Proceedings Fourth European Multigrid conference, Birkhausser, pp. 43-56, (1993).
[14] P. Hemker and H. Schippers, Multiple-grid methods for the solution of Fredhohm integral equations of the second kind, Math. of Comp., Vol. 36, pp. 215-232, (1981).
[15] C. Kelly, A fast multilevel algorithm for integral equations, SIAM J. Num. An., Vol. 32, pp. 501-513, (1995).
[16] R Kress, Linear Integral Equations, Springer-Verlag, (1989).
[17] J. Mandel, On multilevel iterative methods for integral equations of the second kind and related problems, Num. Math., Vol. 46, pp. 147-157, (1985).
[18] C. Pérez, Esquemas Nustrôm Multigrid Paralelos para Ecuaciones Fredholm, de Segunda Especie, M. Sc. Thesis, Universidad de Concepción, ( 1996).
[19] P. Rabinowitz and P. Davis, Methods of Numerical Integration, Academic Press, 2" ed., (1984).
[2] Ph. Anselone. Collectively Compact Operator- Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs. (1971).
[3] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind. Cambridge, University Press (1997).
[4] J. Bramble, Multigrid Methods, Pitman, (1993).
[5] M. Campos and C. Pérez, Paralelización de técnicas para ecuaciones integrales de Fredholm de segunda especie por métodos de cuadratura y multimallas, Departamento de Ingeniería Matemática, Universidad de Concepción, Technical Report 97-09, (1997).
[6] M. Campos and C. Pérez. Tratamiento numérico de ecuaciones integrales mediante técnicas de separación de dominios, Departamento de Ingeniería Matemática, Universidad de Concepción, Technical Report
97-15, (1997).
[7] M. Campos and C. Pérez, On the use of domain separation techniques for solving Fredholm inteqral equations of the second kind, in preparation, ( 1998).
[8] T. Chan and R. Tuminaro, Analysis of a parallel multigrid algorithm., Procceding of the Fourth Copper Mountain conference on multigrid methods, SIAM, pp. 66-86, (1989).
[9] C. Douglas and C. Douglas, A unified convergence theory for abstract multigrid or multilevel methods, serial and parallel, SIAM .J. Num. An., Vol. 30. pp. 136-158, (1993).
[10] C. Douglas and W. Miranker, Constructive interference in parallel algorithms, SIAM J. Num. An., Vol. 25, pp. 376-398, (1988).
[11] I. Goldbcrg, I. Koltracht and P. Lancastcr, Second order parallel algorithms for Fredholm integral equations with continuous displacement kernels, Int. Op. Th., Vol. 10, pp .. 577-594, (1987).
[12] W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, (1985).
[13] W. Hackbusch, The frequency decomposition multi-grid method, Proceedings Fourth European Multigrid conference, Birkhausser, pp. 43-56, (1993).
[14] P. Hemker and H. Schippers, Multiple-grid methods for the solution of Fredhohm integral equations of the second kind, Math. of Comp., Vol. 36, pp. 215-232, (1981).
[15] C. Kelly, A fast multilevel algorithm for integral equations, SIAM J. Num. An., Vol. 32, pp. 501-513, (1995).
[16] R Kress, Linear Integral Equations, Springer-Verlag, (1989).
[17] J. Mandel, On multilevel iterative methods for integral equations of the second kind and related problems, Num. Math., Vol. 46, pp. 147-157, (1985).
[18] C. Pérez, Esquemas Nustrôm Multigrid Paralelos para Ecuaciones Fredholm, de Segunda Especie, M. Sc. Thesis, Universidad de Concepción, ( 1996).
[19] P. Rabinowitz and P. Davis, Methods of Numerical Integration, Academic Press, 2" ed., (1984).
Published
2018-04-04
How to Cite
[1]
C. E. Pérez Wilson, “On the resolution of fredholm integral equations by multigrid domain separation techniques”, Proyecciones (Antofagasta, On line), vol. 17, no. 1, pp. 119-132, Apr. 2018.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
