An approximate solution of a Fredholm integral equation of the first kind by the residual method

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• 作者：V. P. Tanana ; E. Yu. Vishnyakov ; A. I. Sidikova
• 关键词：regularization ; residual method ; continuity module ; error estimation ; ill ; posed problem
• 刊名：Numerical Analysis and Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：9
• 期：1
• 页码：74-81
• 全文大小：515 KB
• 参考文献：1.Kabanikhin, S.I., Obratnye i nekorrektnye zadachi. Uchebnik dlya studentov vysshikh uchebnykh zavedenii (Inverse and Ill-Posed Problems: Textbook for University Students), Novosibirsk: Sibirskoe Nauchnoe Izd-vo, 2009.
  2.Morozov, V.A., Regularization of Ill-Posed Problems and Choice of a Regularization Parameter, Zh. Vych. Mat. Mat. Fiz., 1966, vol. 6, no. 1, pp. 170–175.MATH
  3.Tikhonov, A.N., Regularization of Ill-Posed Problems, Dokl. AN SSSR, 1963, vol. 153, no. 1, pp. 49–52.MathSciNet
  4.Morozov, V.A., The Residual Principle in Solving Operator Equations by the Method of Regularization, Zh. Vych. Mat.Mat. Fiz., 1968, vol. 8, no. 2, pp. 295–309.
  5.Ivanov, V.K., Approximate Solution of an Operator Equations of the First Kind, Zh. Vych. Mat. Mat. Fiz., 1966, vol. 6, no. 6, pp. 1089–1094.MathSciNet
  6.Goncharskii, A.V., Leonov, A.S., and Yagola, A.G., Finite-Difference Approximation of Linear Ill-Posed Problems, Zh. Vych. Mat. Mat. Fiz., 1974, vol. 14, no. 4, pp. 1022–1027.MathSciNet
  7.Vasin, V.V. and Tanana, V.P., Necessary and Sufficient Conditions for Convergence of ProjectionMethods for Linear Unstable Problems, Dokl. AN SSSR, 1974, vol. 215, no. 5, pp. 1032–1034.MathSciNet MATH
  8.Tanana, V.P., Projection Methods and Finite-Difference Approximation of Linear Ill-Posed Problems, Sib. Mat. Zh., 1975, vol. 16, no. 6, pp. 1301–1307.MathSciNet
  9.Vasin, V.V., Discrete Convergence and Finite-Dimensional Approximation of Regularizing Algorithms, Zh. Vych. Mat.Mat. Fiz., 1979, vol. 19, no. 1, pp. 11–21.MathSciNet
  10.Tanana, V.P. and Sidikova, A.I., An Error Estimate of a Regularizing Algorithm Based on the Generalized Residual PrincipleWhen Solving Integral Equations, Vych. Met. Program., 2015, vol. 16, no. 1, pp. 1–9.MathSciNet
  11.Tanana, V.P., On Optimality of Methods for Solving Nonlinear Unstable Problems, Dokl. AN SSSR, 1975, vol. 220, no. 5, pp. 1035–1037.MathSciNet MATH
  12.Tanana, V.P., On the Order-Optimality of the Projection Regularization Method in Solving Inverse Problems, Sib. Zh. Industr. Mat., 2004, vol. 7, no. 2, pp. 117–132.MathSciNet MATH
  
• 作者单位：V. P. Tanana (1)
  E. Yu. Vishnyakov (1)
  A. I. Sidikova (1)
  
  1. South Ural State University, pr. Lenina 76, Chelyabinsk, 630090, Russia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1995-4247

文摘

A Tikhonov finite-dimensional approximation is applied to a Fredholm integral equation of the first kind. This allows using a variational regularization method with a regularization parameter from the residual principle and reducing the problem to a system of linear algebraic equations. The accuracy of the approximate solution is estimated with allowance for the error of the finitedimensional approximation of the problem. The use of this approach is illustrated by solving an inverse boundary value problem for the heat conduction equation.