Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Gradient method for nondensely defined closed unbounded linear operators


Authors: Sung J. Lee and M. Zuhair Nashed
Journal: Proc. Amer. Math. Soc. 88 (1983), 429-435
MSC: Primary 47A50; Secondary 65J10
DOI: https://doi.org/10.1090/S0002-9939-1983-0699408-9
MathSciNet review: 699408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper establishes the convergence of the steepest descent method for least-squares solutions of operator equations in Hilbert spaces for any (nondensely defined, unbounded) closed linear operator with closed range. This is done by using a graph topology, an explicit graph topology adjoint, and existing theory of steepest descent for bounded linear operators.


References [Enhancements On Off] (What's this?)

  • [1] R. Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9-23. MR 0123188 (23:A517)
  • [2] E. A. Coddington and A. Dijksma, Adjoint subspaces in Banach spaces, with application to ordinary differential subspaces, Ann. Mat. Pura. Appl. 118 (1978), 1-118. MR 533601 (81k:47036)
  • [3] M. R. Hestenes, Relative self-adjoint operators in Hilbert spaces, Pacific J. Math. 11 (1961), 1315-1357. MR 0136996 (25:456)
  • [4] J. W. Jerome and L. L. Schumaker, On $ Lg$-splines, J. Approx. Theory 2 (1969), 29-49. MR 0241864 (39:3201)
  • [5] W. J. Kammerer and M. Z. Nashed, Steepest descent for singular linear operators with nonclosed range, Applicable Anal. 1 (1971), 143-159. MR 0290134 (44:7319)
  • [6] -, 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)
  • [7] L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, Pergamon Press, London and New York, 1964. MR 0213845 (35:4699)
  • [8] L. J. Lardy, A series representation for generalized inverse of a closed linear operator, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 152-157. MR 0473881 (57:13540)
  • [9] S. J. Lee, Boundary conditions for linear manifolds, I, J. Math. Anal. Appl. 73 (1980), 366-380. MR 563989 (84d:47003a)
  • [10] S. J. Lee and M. Z. Nashed, Operator parts and generalized inverses of multi-valued operators, with applications to ordinary differential subspaces (to appear).
  • [11] -, Least-squares solutions of multi-valued linear operators, J. Approx. Theory (to appear).
  • [12] J. Locker, Weak steepest descent for linear boundary value problems, Indiana Univ. Math. J. 25 (1976), 525-530. MR 0418456 (54:6495)
  • [13] T. R. Lucas, A generalization of $ L$-splines, Numer. Math. 15 (1970), 359-370. MR 0269080 (42:3976)
  • [14] S. F. McCormick and G. H. Rodrigue, A uniform approach to gradient methods for linear operator equations, J. Math. Anal. Appl. 49 (1975), 275-285. MR 0377564 (51:13735)
  • [15] M. Z. Nashed, Steepest descent for singular linear operator equations, SIAM J. Numer. Anal. 7 (1970), 358-362. MR 0269093 (42:3989)
  • [16] -, Perturbations and approximations for generalized inverses and linear operator equations, Generalized Inverses and Applications (M. Z. Nashed, ed.), Academic Press, New York, 1976, pp. 325-396. MR 0500249 (58:17923)
  • [17] W. V. Petryshyn, Direct and iterative methods for the solution of linear operator equations in Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 136-175. MR 0145651 (26:3180)
  • [18] -, On generalized inverses and uniform convergence of $ {(I - \beta K)^n}$ with applications to iterative methods, J. Math. Anal. Appl. 18 (1967), 417-439. MR 0208381 (34:8191)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A50, 65J10

Retrieve articles in all journals with MSC: 47A50, 65J10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699408-9
Keywords: Graph topology, graph topology adjoint, operator part, gradient method, steepest descent, unbounded linear operator, normal equations, iterative methods, generalized inverse of subspace
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society