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

Marti, J. T.

doi:10.1090/S0025-5718-1980-0559200-8

On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems
HTML articles powered by AMS MathViewer

by J. T. Marti PDF

Math. Comp. 34 (1980), 521-527 Request permission

Abstract:

The paper studies a finite element algorithm giving approximations to the minimum-norm solution of ill-posed problems of the form $Af = g$, 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 $A{A^\ast }$. As an example, the method has been applied to the problem of evaluating the second derivative f of a function g numerically.

References

Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
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
Joel N. Franklin, On Tikhonov’s method for ill-posed problems, Math. Comp. 28 (1974), 889–907. MR 375817, DOI 10.1090/S0025-5718-1974-0375817-5
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, DOI 10.1007/BF01411844
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 320677, DOI 10.1016/0022-247X(72)90002-9
Jürg T. Marti, Konvexe Analysis, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 54, Birkhäuser Verlag, Basel-Stuttgart, 1977 (German). MR 0511737

On the Numerical Computation of Minimum Norm Solutions of Fredholm Integral Equations of the First Kind Having a Symmetric Kernel

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, DOI 10.1137/0715071
Martin H. Schultz, Spline analysis, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0362832

Soviet Math. Dokl.

Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, 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. MR 0455365
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 181129, DOI 10.1016/0016-0032(65)90209-7
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 345436, DOI 10.1016/0016-0032(74)90103-3