On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems

Author:
J. T. Marti

Journal:
Math. Comp. **34** (1980), 521-527

MSC:
Primary 65J10; Secondary 47A50

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559200-8

MathSciNet review:
559200

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Abstract | References | Similar Articles | Additional Information

Abstract: The paper studies a finite element algorithm giving approximations to the minimum-norm solution of ill-posed problems of the form , where *A* is a bounded linear operator from one Hubert space to another. It is shown that the algorithm is norm convergent in the general case and an error bound is derived for the case where *g* is in the range of . As an example, the method has been applied to the problem of evaluating the second derivative *f* of a function *g* numerically.

**[1]**Philip M. Anselone,*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR**0443383****[2]**Kendall E. Atkinson,*A survey of numerical methods for the solution of Fredholm integral equations of the second kind*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR**0483585****[3]**Joel N. Franklin,*On Tikhonov’s method for ill-posed problems*, Math. Comp.**28**(1974), 889–907. MR**0375817**, https://doi.org/10.1090/S0025-5718-1974-0375817-5**[4]**Jack Graves and P. M. Prenter,*Numerical iterative filters applied to first kind Fredholm integral equations*, Numer. Math.**30**(1978), no. 3, 281–299. MR**502805**, https://doi.org/10.1007/BF01411844**[5]**W. J. Kammerer and M. Z. Nashed,*Iterative methods for best approximate solutions of linear integral equations of the first and second kinds*, J. Math. Anal. Appl.**40**(1972), 547–573. MR**0320677**, https://doi.org/10.1016/0022-247X(72)90002-9**[6]**Jürg T. Marti,*Konvexe Analysis*, Birkhäuser Verlag, Basel-Stuttgart, 1977 (German). Lehrbücher und Monographien aus dem Gebiet der Exakten Wissenschaften, Mathematische Reihe, Band 54. MR**0511737****[7]**J. T. MARTI,*On the Numerical Computation of Minimum Norm Solutions of Fredholm Integral Equations of the First Kind Having a Symmetric Kernel*, Report 78-01, Seminar für Angew. Math., ETH, Zurich, 1978.**[8]**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**[9]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832****[10]**A. N. TIKHONOV, ``Solution of incorrectly formulated problems and the regularization method,''*Soviet Math. Dokl.*, v. 4, 1963, pp. 1035-1038.**[11]**A. N. TIKHONOV, ``Regularizaron of incorrectly posed problems,''*Soviet Math. Dokl.*, v. 4, 1963, pp. 1624-1627.**[12]**Andrey N. Tikhonov and Vasiliy Y. Arsenin,*Solutions of ill-posed problems*, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR**0455365****[13]**S. Twomey,*The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements*, J. Franklin Inst.**279**(1965), 95–109. MR**0181129**, https://doi.org/10.1016/0016-0032(65)90209-7**[14]**V. Vemuri and Fang Pai Chen,*An initial value method for solving Fredholm integral equation of the first kind*, J. Franklin Inst.**297**(1974), 187–200. MR**0345436**, https://doi.org/10.1016/0016-0032(74)90103-3

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559200-8

Keywords:
Ill-posed problem,
algorithm,
minimum-norm solution,
Fredholm integral equation of the first kind

Article copyright:
© Copyright 1980
American Mathematical Society