An improved version of Marti's method for solving ill-posed linear integral equations

Authors:
Heinz W. Engl and Andreas Neubauer

Journal:
Math. Comp. **45** (1985), 405-416

MSC:
Primary 65R20; Secondary 45L10

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804932-1

MathSciNet review:
804932

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Abstract | References | Similar Articles | Additional Information

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).

**[1]**H. W. Engl,*On the convergence of regularization methods for ill-posed linear operator equations*, Improperly posed problems and their numerical treatment (Oberwolfach, 1982) Internat. Schriftenreihe Numer. Math., vol. 63, Birkhäuser, Basel, 1983, pp. 81–95. MR**726766****[2]**H. W. Engl,*Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates*, J. Optim. Theory Appl.**52**(1987), no. 2, 209–215. MR**879198**, https://doi.org/10.1007/BF00941281**[3]**Heinz W. Engl and Andreas Neubauer,*Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind*, Constructive methods for the practical treatment of integral equations (Oberwolfach, 1984) Internat. Schriftenreihe Numer. Math., vol. 73, Birkhäuser, Basel, 1985, pp. 120–141. MR**882562****[4]**C. W. Groetsch, "The parameter choice problem in linear regularization," in*Ill-Posed Problems, Theory and Practise*(M. Z. Nashed, ed.). (To appear.)**[5]**C. W. Groetsch,*Comments on Morozov’s discrepancy principle*, Improperly posed problems and their numerical treatment (Oberwolfach, 1982) Internat. Schriftenreihe Numer. Math., vol. 63, Birkhäuser, Basel, 1983, pp. 97–104. MR**726767****[6]**C. W. Groetsch, J. T. King, and D. Murio,*Asymptotic analysis of a finite element method for Fredholm equations of the first kind*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 1–11. MR**755337****[7]**C. W. Groetsch,*The theory of Tikhonov regularization for Fredholm equations of the first kind*, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**742928****[8]**Kevin R. Hickey and Glenn R. Luecke,*Remarks on Marti’s method for solving first kind equations*, SIAM J. Numer. Anal.**19**(1982), no. 3, 623–628. MR**656476**, https://doi.org/10.1137/0719043**[9]**J. T. Marti,*An algorithm for computing minimum norm solutions of Fredholm integral equations of the first kind*, SIAM J. Numer. Anal.**15**(1978), no. 6, 1071–1076. MR**512683**, https://doi.org/10.1137/0715071**[10]**J. T. Marti,*On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems*, Math. Comp.**34**(1980), no. 150, 521–527. MR**559200**, https://doi.org/10.1090/S0025-5718-1980-0559200-8**[11]**J. T. Marti,*On a regularization method for Fredholm equations of the first kind using Sobolev spaces*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 59–66. MR**755342****[12]**M. Zuhair Nashed (ed.),*Generalized inverses and applications*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. University of Wisconsin, Mathematics Research Center, Publication No. 32. MR**0451661****[13]**V. A. Morozov,*On the solution of functional equations by the method of regularization*, Soviet Math. Dokl.**7**(1966), 414–417. MR**0208819****[14]**A. N. Tikhonov & V. Y. Arsenin,*Solution of Ill-Posed Problems*, English transl., Wiley, New York, 1977.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804932-1

Keywords:
Ill-posed problems,
regularization methods,
integral equations of the first kind

Article copyright:
© Copyright 1985
American Mathematical Society