A collocation method for Fredholm integral equations of the first kind via iterative regularization scheme
Abstract
To solve the ill-posed integral equations, we use the regularized collocation method. This numerical method is a combination of the Legendre polynomials with non-stationary iterated Tikhonov regularization with fixed parameter. A theoretical justification of the proposed method under the required assumptions is detailed. Finally, numerical experiments demonstrate the efficiency of this method.
Keyword : ill-posed problems, iterative regularization scheme, Legendre collocation method, integral equations of the first kind
How to Cite
Bechouat, T. (2023). A collocation method for Fredholm integral equations of the first kind via iterative regularization scheme. Mathematical Modelling and Analysis, 28(2), 237–254. https://doi.org/10.3846/mma.2023.16453
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S. Yang, X. Luo, F. Li and G. Long. A fast multiscale Galerkin method for the first kind ill-posed integral equations via iterated regularization. Applied Mathematics and Computation, 219(21):10527–10537, 2013. https://doi.org/10.1016/j.amc.2013.04.029
Z. Chen, S. Cheng, G. Nelakanti and H. Yang. A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization. International Journal of Computer Mathematics, 87(3):565–582, 2010. https://doi.org/10.1080/00207160802155302
M. Donatelli. On nondecreasing sequences of regularization parameters for nonstationary iterated Tikhonov. Numerical Algorithms, 60(4):651–668, 2012.
H.W. Engl, M. Hanke and A. Neubauer. Regularization of inverse problems, volume 375. Springer Science & Business Media, 1996.
L. Fanchun, Y. Suhua, L. Xingjun and P. Yubing. Multilevel iterative algorithm for solving Fredholm integral equation of the first kind. Mathematica Numerica Sinica, 35(3):225–238, 2013. https://doi.org/10.12286/jssx.2013.3.225
U. Hamarik, E. Avi and A. Ganina. On the solution of ill-posed problems by projection methods with a posteriori choice of the discretization level. Mathematical Modelling and Analysis, 7(2):241–252, 2002. https://doi.org/10.3846/13926292.2002.9637196
M. Hanke and C.W. Groetsch. Nonstationary iterated Tikhonov regularization. Journal of Optimization Theory and Applications, 98(1):37–53, 1998. https://doi.org/10.1023/A:1022680629327
G. Huang, L. Reichel and F. Yin. Projected nonstationary iterated Tikhonov regularization. BIT Numerical Mathematics, 56(2):467–487, 2016. https://doi.org/10.1007/s10543-015-0568-7
A.V. Kryanev. An iterative method for solving incorrectly posed problems. USSR Computational Mathematics and Mathematical Physics, 14(1):24–35, 1974. https://doi.org/10.1016/0041-5553(74)90133-5
S. Lu and S.V. Pereverzev. Regularization theory for ill-posed problems. de Gruyter, Berlin, Boston, 2013. https://doi.org/10.1515/9783110286496
Y. Lu, L. Shen and Y. Xu. Multi-parameter regularization methods for high-resolution image reconstruction with displacement errors. IEEE Transactions on Circuits and Systems I: Regular Papers, 54(8):1788–1799, 2007. https://doi.org/10.1109/TCSI.2007.902535
X. Luo, L. Fan, Y. Wu and F. Li. Fast multi-level iteration methods with compression technique for solving ill-posed integral equations. Journal of Computational and Applied Mathematics, 256:131–151, 2014. https://doi.org/10.1109/TCSI.2007.902535
M.T. Nair. Linear operator equations: approximation and regularization. World Scientific, 2009.
M.T. Nair. Quadrature based collocation methods for integral equations of the first kind. Advances in Computational Mathematics, 36(2):315–329, 2012. https://doi.org/10.1007/s10444-011-9196-1
M.T. Nair and S.V. Pereverzev. Regularized collocation method for Fredholm integral equations of the first kind. Journal of Complexity, 23(4):454–467, 2007. https://doi.org/10.1016/j.jco.2006.09.002
B. Neggal, N. Boussetila and F. Rebbani. Projected Tikhonov regularization method for Fredholm integral equations of the first kind. Journal of Inequalities and Applications, 2016(195):1–21, 2016. https://doi.org/10.1186/s13660-016-1137-6
B. Nemati Saray. Sparse multiscale representation of Galerkin method for solving linear-mixed Volterra-Fredholm integral equations. Mathematical Methods in the Applied Sciences, 43(5):2601–2614, 2020. https://doi.org/doi.org/10.1002/mma.6068
T. Poggio and C.R. Shelton. On the mathematical foundations of learning. Bulletin of the American Mathematical Society, 39(1):1–49, 2002. https://doi.org/10.1090/S0273-0979-01-00923-5
A. Quarteroni and A. Valli. Numerical approximation of partial differential equations, volume 23. Springer Science & Business Media, 2008.
M.P. Rajan. A modified convergence analysis for solving Fredholm integral equations of the first kind. Integral Equations and Operator Theory, 49(4):511– 516, 2004. https://doi.org/10.1007/s00020-002-1213-9
B. Tahar, B. Nadjib and R. Faouzia. A variant of projection regularization method for ill-posed linear operator equations. International Journal of Computational Methods, 18(04):2150008, 2021. https://doi.org/10.1142/S0219876221500080
J. Thomas King and D. Chillingworth. Approximation of generalized inverses by iterated regularization. Numerical Functional Analysis and Optimization, 1(5):499–513, 1979. https://doi.org/10.1080/01630567908816031
C.R. Vogel and M.E. Oman. Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE transactions on image processing, 7(6):813–824, 1998. https://doi.org/10.1109/83.679423
S. Yang, X. Luo, F. Li and G. Long. A fast multiscale Galerkin method for the first kind ill-posed integral equations via iterated regularization. Applied Mathematics and Computation, 219(21):10527–10537, 2013. https://doi.org/10.1016/j.amc.2013.04.029