Gradient method for nondensely defined closed unbounded linear operators
Authors:
Sung J. Lee and M. Zuhair Nashed
Journal:
Proc. Amer. Math. Soc. 88 (1983), 429435
MSC:
Primary 47A50; Secondary 65J10
MathSciNet review:
699408
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Abstract: The paper establishes the convergence of the steepest descent method for leastsquares 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.
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 [1]
 R. Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 923. 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), 1118. MR 533601 (81k:47036)
 [3]
 M. R. Hestenes, Relative selfadjoint operators in Hilbert spaces, Pacific J. Math. 11 (1961), 13151357. MR 0136996 (25:456)
 [4]
 J. W. Jerome and L. L. Schumaker, On splines, J. Approx. Theory 2 (1969), 2949. 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), 143159. 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), 547573. 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), 152157. MR 0473881 (57:13540)
 [9]
 S. J. Lee, Boundary conditions for linear manifolds, I, J. Math. Anal. Appl. 73 (1980), 366380. MR 563989 (84d:47003a)
 [10]
 S. J. Lee and M. Z. Nashed, Operator parts and generalized inverses of multivalued operators, with applications to ordinary differential subspaces (to appear).
 [11]
 , Leastsquares solutions of multivalued linear operators, J. Approx. Theory (to appear).
 [12]
 J. Locker, Weak steepest descent for linear boundary value problems, Indiana Univ. Math. J. 25 (1976), 525530. MR 0418456 (54:6495)
 [13]
 T. R. Lucas, A generalization of splines, Numer. Math. 15 (1970), 359370. 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), 275285. MR 0377564 (51:13735)
 [15]
 M. Z. Nashed, Steepest descent for singular linear operator equations, SIAM J. Numer. Anal. 7 (1970), 358362. 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. 325396. 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), 136175. MR 0145651 (26:3180)
 [18]
 , On generalized inverses and uniform convergence of with applications to iterative methods, J. Math. Anal. Appl. 18 (1967), 417439. MR 0208381 (34:8191)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198306994089
PII:
S 00029939(1983)06994089
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
