Fast solvers of weakly singular integral equations of the second kind
Abstract
We discuss the bounds of fast solving weakly singular Fredholm integral equations of the second kind with a possible diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of sample values. In this situation, a fast/quasifast solver is constructed. Thus the complexity of weakly singular integral equations occurs to be close to that of equations with smooth data without singularities. Our construction of fast/quasifast solvers is based on the periodization of the problem.
Keyword : Complexity, Fast solvers, Weakly singular integral equations, Fredholm equations, Periodization, Trigonometric collocation
How to Cite
Rehman, S., Pedas, A., & Vainikko, G. (2018). Fast solvers of weakly singular integral equations of the second kind. Mathematical Modelling and Analysis, 23(4), 639-664. https://doi.org/10.3846/mma.2018.039
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Pedas and G. Vainikko. Integral equations with diagonal and boundary singularities of the kernel. ZAA, 25:487–516,2006.https://doi.org/10.4171/ZAA/1304
A. Pedas and G. Vainikko. Smoothing transformation and piece-wise polynomial projection methods for weakly singular Fredholm integral equations. Comm. Pure and Appl. Anal, 5:395–413, 2006.https://doi.org/10.3934/cpaa.2006.5.395
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L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, Philadelphia, 1997. https://doi.org/10.1137/1.9780898719574
E. Vainikko and G. Vainikko. A spline product quasi-interpolation method for weakly singular Fredholm integral equations. SIAM J. Numer. Anal, 46:1799–1820, 2008. https://doi.org/10.1137/070693308
G. Vainikko. Multidimensional Weakly Singular Integral Equations. Springer-Verag, Berlin, 1993.https://doi.org/10.1007/BFb0088979
G. Vainikko. GMRES and discrete approximation of operators. Proc. Estonian Acad Sci. Phys. Math,53:124–131, 2004.
G. Vainikko. Fast solvers of integral equations of the second kind: quadrature methods. J. Integr. Eq. Appl., 17:91–120, 2005. https://doi.org/10.1216/jiea/1181075312
G. Vainikko. Fast solvers of integral equations. Lecture notes, HUT, UT, 2006.http://www.ut.ee/ ̃ gen/FASTlecturesSIAM.pdf
G. Vainikko. Fast wavelet solvers of periodic integral equations. J. Analysis, 14:243–265, 2006
G. Vainikko, A. Kivinukk and J. Lippus. Fast solvers of integral equations of the second kind: wavelet methods. J. Complexity, 21:243–273, 2005. https://doi.org/10.1016/j.jco.2004.07.002
G. Vainikko and A. Pedas. The properties of solutions of weakly singular integral equations. J. Austral. Math. Soc, Ser. B, 22:419–430, 1980.
G. Vainikko and I. Zolk. Fast spline quasicollocation solvers of integral equations.Math. Modelling and Analysis,12:515–538, 2007. https://doi.org/10.3846/1392-6292.2007.12.515-538
A. G. Werschulz. Where does smoothness count the most for Fredholm equations of the second kind with noisy information?J. Complexity, 19:758–798, 2003.https://doi.org/10.1016/S0885-064X(03)00030-X
A. Zygmund. Trigonometric Series, volume 1, 2. Cambridge Univ. Press, 1959.
W. Dahmen, H. Harbrecht and R. Schneider. Compression techniques for boundary integral equations asymptotically optimal complexity estimates . SIAMJ . Numer . Anal, 43:2251–2271, 2006.https://doi.org/10.1137/S0036142903428852
T. Diogo, P. M. Lima, A. Pedas and G. Vainikko. Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations. Appl. Numer. Math, 114:63–76, 2017.
K. V. Emelyanov and A. M. Il’in. The number of arithmetical operations necessary for the approximate solution of Fredholm integral equations. USSR Comput. Math. Phys, 7:259–267, 1967. https://doi.org/10.1016/0041-5553(67)90160-7(In Russian)
K. Frank, S. Heinrich and S. Pereverzev. Information complexity of multivariate Fredholm integral equations in Sobolev classes. J. Complexity, 12:17–34, 1996. https://doi.org/10.1006/jcom.1996.0004
I. G. Graham. Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels.J. Integral Equations,4:1–30, 1982.
M. Kolk, A. Pedas and G. Vainikko. High-order methods for Volterra integral equations wit general weak singularities. Numer. Funct. Anal. Optim, 30(10):1002–1024, 2009.https://doi.org/10.1080/01630560903393154
G. Monegato and L. Scuderi. High order methods for weakly singular integralequations with nonsmooth input functions. Math. Comput, 67:1493–1515, 1998.https://doi.org/10.1090/S0025-5718-98-01005-9
O. Nevanlinna. Convergence of iterations for linear equations. Birkhauser, Basel,1993. https://doi.org/10.1007/978-3-0348-8547-8
K. Orav-Puurand, A. Pedas and G. Vainikko. Central part interpolation schemes for integral equations with singularities. J. Int. Eq. Appl, 29(3):401–440, 2017.https://doi.org/10.1216/JIE-2017-29-3-401
I. Parts, A. Pedas and E. Tamme. Piecewise polynomial collocation for Fredholm integro-differential equations with weakly singular kernels.SIAM J. Numer. Anal, 43:1897–1911, 2005.https://doi.org/10.1137/040612452
A. Pedas and G. Vainikko. Integral equations with diagonal and boundary singularities of the kernel. ZAA, 25:487–516,2006.https://doi.org/10.4171/ZAA/1304
A. Pedas and G. Vainikko. Smoothing transformation and piece-wise polynomial projection methods for weakly singular Fredholm integral equations. Comm. Pure and Appl. Anal, 5:395–413, 2006.https://doi.org/10.3934/cpaa.2006.5.395
A. Pedas and G. Vainikko. What is the complexity of periodic weakly singular integral equations? BIT Num. Math, 48:315–335, 2008. https://doi.org/10.1007/s10543-008-0181-0
A. Pedas and G. Vainikko. On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals. J. Comp. Appl. Math, 229:440–451, 2009.https://doi.org/10.1016/j.cam.2008.04.009
S. V. Pereverzev. On the complexity of the problem of finding the solutions of Fredholm equations of the second kind with smooth kernels. I. Ukrain. Math. Zh, 40(1):71–76, 1988. https://doi.org/10.1007/bf01056451
S. V. Pereverzev. On the complexity of the problem of finding the solutions of Fredholm equations of the second kind with smooth kernels. II. Ukrain. Math. Zh, 41(2):169–173, 1989.https://doi.org/10.1007/BF01060382
S. V. Pereverzev. Hyperbolic cross and the complexity of the approximate solution of Fredholm integral equations of the second kind with differentiable kernels. Sibirsk. Math. Zh, 32(1):85–92, 1991. https://doi.org/10.1007/BF00970164
S. V. Pereverzev and K. Sh. Makhkamov. Information complexity of weakly singular integral equations. Ukrainian Math. J, 46:1527–1533, 1994. https://doi.org/10.1007/BF01058886
J. Pitkaranta. Estimates for the derivatives of solutions to weakly singular Fredholm integral equations.SIAM J. Math. Anal,11:952–968, 1980. https://doi.org/10.1137/0511085
J. Saranen and G. Vainikko.Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer, Berlin, 2002. https://doi.org/10.1007/978-3-662-04796-5
J. F. Traub, G. Wozniakowski and H. Wozniakowski. A General Theory of Optimal Algorithms. Academic Press, New York, 1980
J. F. Traub, G. Wozniakowski and H. Wozniakowski. Information-Based Complexity. American Press, Boston, 1988.
L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, Philadelphia, 1997. https://doi.org/10.1137/1.9780898719574
E. Vainikko and G. Vainikko. A spline product quasi-interpolation method for weakly singular Fredholm integral equations. SIAM J. Numer. Anal, 46:1799–1820, 2008. https://doi.org/10.1137/070693308
G. Vainikko. Multidimensional Weakly Singular Integral Equations. Springer-Verag, Berlin, 1993.https://doi.org/10.1007/BFb0088979
G. Vainikko. GMRES and discrete approximation of operators. Proc. Estonian Acad Sci. Phys. Math,53:124–131, 2004.
G. Vainikko. Fast solvers of integral equations of the second kind: quadrature methods. J. Integr. Eq. Appl., 17:91–120, 2005. https://doi.org/10.1216/jiea/1181075312
G. Vainikko. Fast solvers of integral equations. Lecture notes, HUT, UT, 2006.http://www.ut.ee/ ̃ gen/FASTlecturesSIAM.pdf
G. Vainikko. Fast wavelet solvers of periodic integral equations. J. Analysis, 14:243–265, 2006
G. Vainikko, A. Kivinukk and J. Lippus. Fast solvers of integral equations of the second kind: wavelet methods. J. Complexity, 21:243–273, 2005. https://doi.org/10.1016/j.jco.2004.07.002
G. Vainikko and A. Pedas. The properties of solutions of weakly singular integral equations. J. Austral. Math. Soc, Ser. B, 22:419–430, 1980.
G. Vainikko and I. Zolk. Fast spline quasicollocation solvers of integral equations.Math. Modelling and Analysis,12:515–538, 2007. https://doi.org/10.3846/1392-6292.2007.12.515-538
A. G. Werschulz. Where does smoothness count the most for Fredholm equations of the second kind with noisy information?J. Complexity, 19:758–798, 2003.https://doi.org/10.1016/S0885-064X(03)00030-X
A. Zygmund. Trigonometric Series, volume 1, 2. Cambridge Univ. Press, 1959.