On the convergence of an algorithm computing minimumnorm solutions of illposed problems
Author:
J. T. Marti
Journal:
Math. Comp. 34 (1980), 521527
MSC:
Primary 65J10; Secondary 47A50
MathSciNet review:
559200
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The paper studies a finite element algorithm giving approximations to the minimumnorm solution of illposed 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, PrenticeHall Inc., Englewood
Cliffs, N. J., 1971. With an appendix by Joel Davis; PrenticeHall Series
in Automatic Computation. MR 0443383
(56 #1753)
 [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
(58 #3577)
 [3]
Joel
N. Franklin, On Tikhonov’s method for
illposed problems, Math. Comp. 28 (1974), 889–907. MR 0375817
(51 #12007), http://dx.doi.org/10.1090/S00255718197403758175
 [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
(81f:65091), http://dx.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
(47 #9213)
 [6]
Jürg
T. Marti, Konvexe Analysis, Birkhäuser Verlag, Basel,
1977 (German). Lehrbücher und Monographien aus dem Gebiet der Exakten
Wissenschaften, Mathematische Reihe, Band 54. MR 0511737
(58 #23497)
 [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 7801, 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
(80b:65154), http://dx.doi.org/10.1137/0715071
 [9]
Martin
H. Schultz, Spline analysis, PrenticeHall Inc., Englewood
Cliffs, N.J., 1973. PrenticeHall Series in Automatic Computation. MR 0362832
(50 #15270)
 [10]
A. N. TIKHONOV, ``Solution of incorrectly formulated problems and the regularization method,'' Soviet Math. Dokl., v. 4, 1963, pp. 10351038.
 [11]
A. N. TIKHONOV, ``Regularizaron of incorrectly posed problems,'' Soviet Math. Dokl., v. 4, 1963, pp. 16241627.
 [12]
Andrey
N. Tikhonov and Vasiliy
Y. Arsenin, Solutions of illposed problems, V. H. Winston
& Sons, Washington, D.C.: John Wiley & Sons, New York, 1977.
Translated from the Russian; Preface by translation editor Fritz John;
Scripta Series in Mathematics. MR 0455365
(56 #13604)
 [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
(31 #5358)
 [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
(49 #10172)
 [1]
 P. M. ANSELONE, Collectively Compact Operator Approximation Theory, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
 [2]
 K. E. ATKINSON, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
 [3]
 J. N. FRANKLIN, ``On Tikhonov's method for illposed problems,'' Math. Comp., v. 28, 1974, pp. 889907. MR 0375817 (51:12007)
 [4]
 J. GRAVES & P. M. PRENTER, ``Numerical iterative filters applied to first kind Fredholm integral equations,'' Numer. Math., v. 30, 1978, pp. 281299. MR 502805 (81f:65091)
 [5]
 W. J. KAMMERER & M. Z. NASHED, ``Iterative methods for best approximate solutions of linear integral equations of the first and second kinds,'' J. Math. Anal. Appl., v. 40, 1972, pp. 547573. MR 0320677 (47:9213)
 [6]
 J. T. MARTI, Konvexe Analysis, Birkhäuser Verlag, Basel, 1977. MR 0511737 (58:23497)
 [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 7801, 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., v. 15, 1978, pp. 10711076. MR 512683 (80b:65154)
 [9]
 M. H. SCHULTZ, Spline Analysis, PrenticeHall, Englewood Cliffs, N. J., 1973. MR 0362832 (50:15270)
 [10]
 A. N. TIKHONOV, ``Solution of incorrectly formulated problems and the regularization method,'' Soviet Math. Dokl., v. 4, 1963, pp. 10351038.
 [11]
 A. N. TIKHONOV, ``Regularizaron of incorrectly posed problems,'' Soviet Math. Dokl., v. 4, 1963, pp. 16241627.
 [12]
 A. N. TIKHONOV & V. Y. ARSENIN, Solutions of IllPosed Problems, Winston, Washington, D. C., 1977. MR 0455365 (56:13604)
 [13]
 S. TWOMEY, ``The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,'' J. Franklin Inst., v. 279, 1965, pp. 95105. MR 0181129 (31:5358)
 [14]
 V. VEMURI & FANGPAI CHEN, ``An initial value method for solving Fredholm integral equations of the first kind,'' J. Franklin Inst., v. 297, 1974, pp. 187200. MR 0345436 (49:10172)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65J10,
47A50
Retrieve articles in all journals
with MSC:
65J10,
47A50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005592008
PII:
S 00255718(1980)05592008
Keywords:
Illposed problem,
algorithm,
minimumnorm solution,
Fredholm integral equation of the first kind
Article copyright:
© Copyright 1980 American Mathematical Society
