An improved version of Marti's method for solving illposed linear integral equations
Authors:
Heinz W. Engl and Andreas Neubauer
Journal:
Math. Comp. 45 (1985), 405416
MSC:
Primary 65R20; Secondary 45L10
MathSciNet review:
804932
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Abstract 
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Abstract: We propose an algorithm for solving linear integral equations of the first kind that can be viewed as a variant of Marti's method; as opposed to that method, our algorithm leads to optimal convergence rates (also with noisy data).
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 [1]
 H. W. Engl, "On the convergence of regularization methods for illposed linear operator equations," in Improperly Posed Problems and Their Numerical Treatment (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1983, pp. 8195. MR 726766 (85e:65026)
 [2]
 H. W. Engl, "Discrepancy principles for Tikhonov regularization of illposed problems leading to optimal convergence rates," J. Optim. Theory Appl. (To appear.) MR 879198 (88b:49045)
 [3]
 H. W. Engl & A. Neubauer, "Optimal discrepancy principles for the Tikhonovregularization of integral equations of the first kind," in Constructive Methods for the Practical Treatment of Integral Equations (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1985, pp. 120141. MR 882562
 [4]
 C. W. Groetsch, "The parameter choice problem in linear regularization," in IllPosed Problems, Theory and Practise (M. Z. Nashed, ed.). (To appear.)
 [5]
 C. W. Groetsch, "Comments on Morozov's discrepancy principle," in Improperly Posed Problems and Their Numerical Treatment (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1983, pp. 97104. MR 726767 (85d:65036)
 [6]
 C. W. Groetsch, J. T. King & D. Murio, "Asymptotic analysis of a finite element method for Fredholm equations of the first kind," in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and G. F. Miller, eds.), Academic Press, London, 1982, pp. 111. MR 755337 (85k:65107)
 [7]
 C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984. MR 742928 (85k:45020)
 [8]
 K. Hickey & G. Luecke, "Remarks on Marti's method for solving first kind equations," SIAM J. Numer. Anal., v. 19, 1982, pp. 623628. MR 656476 (83h:47007)
 [9]
 J. T. Marti, "An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind," SIAM J. Numer. Anal., v. 15, 1978, pp. 10711076. MR 512683 (80b:65154)
 [10]
 J. T. Marti, "On the convergence of an algorithm computing minimumnorm solutions of illposed problems," Math. Comp., v. 34, 1980, pp. 521527. MR 559200 (83g:65058)
 [11]
 J. T. Marti, "On a regularization method for Fredholm equations of the first kind using Sobolev spaces," in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and G. F. Miller, eds.), Academic Press, London, 1982, pp. 5966. MR 755342
 [12]
 M. Z. Nashed (ed.), Generalized Inverses and Applications, Academic Press, New York, 1976. MR 0451661 (56:9943)
 [13]
 A. Morozov, "On the solution of functional equations by the method of regularization," Soviet Math. Dokl., v. 7, 1966, pp. 414417. MR 0208819 (34:8628)
 [14]
 A. N. Tikhonov & V. Y. Arsenin, Solution of IllPosed Problems, English transl., Wiley, New York, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049321
PII:
S 00255718(1985)08049321
Keywords:
Illposed problems,
regularization methods,
integral equations of the first kind
Article copyright:
© Copyright 1985
American Mathematical Society
