Gradient method for nondensely defined closed unbounded linear operators

Lee, Sung J.; Nashed, M. Zuhair

doi:10.1090/S0002-9939-1983-0699408-9

Gradient method for nondensely defined closed unbounded linear operators
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by Sung J. Lee and M. Zuhair Nashed PDF

Proc. Amer. Math. Soc. 88 (1983), 429-435 Request permission

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

Richard Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9–23. MR 123188
Earl A. Coddington and Aalt Dijksma, Adjoint subspaces in Banach spaces, with applications to ordinary differential subspaces, Ann. Mat. Pura Appl. (4) 118 (1978), 1–118. MR 533601, DOI 10.1007/BF02415124
Magnus R. Hestenes, Relative self-adjoint operators in Hilbert space, Pacific J. Math. 11 (1961), 1315–1357. MR 136996
J. W. Jerome and L. L. Schumaker, On $Lg$-splines, J. Approximation Theory 2 (1969), 29–49. MR 241864, DOI 10.1016/0021-9045(69)90029-x
W. J. Kammerer and M. Z. Nashed, Steepest descent for singular linear operators with nonclosed range, Applicable Anal. 1 (1971), no. 2, 143–159. MR 290134, DOI 10.1080/00036817108839011
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
L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 46, The Macmillan Company, New York, 1964. Translated from the Russian by D. E. Brown; Edited by A. P. Robertson. MR 0213845
Lawrence J. Lardy, A series representation for the generalized inverse of a closed linear operator, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 2, 152–157 (English, with Italian summary). MR 473881
Sung J. Lee, Boundary conditions for linear manifolds. I, J. Math. Anal. Appl. 73 (1980), no. 2, 336–380. MR 563989, DOI 10.1016/0022-247X(80)90284-X

Operator parts and generalized inverses of multi-valued operators, with applications to ordinary differential subspaces

Least-squares solutions of multi-valued linear operators

John Locker, Weak steepest descent for linear boundary value problems, Indiana Univ. Math. J. 25 (1976), no. 6, 525–530. MR 418456, DOI 10.1512/iumj.1976.25.25041
Th. R. Lucas, A generalization of $L$-splines, Numer. Math. 15 (1970), 359–370. MR 269080, DOI 10.1007/BF02165507
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 377564, DOI 10.1016/0022-247X(75)90179-1
M. Z. Nashed, Steepest descent for singular linear operator equations, SIAM J. Numer. Anal. 7 (1970), 358–362. MR 269093, DOI 10.1137/0707027
M. Zuhair Nashed, Perturbations and approximations for generalized inverses and linear operator equations, Generalized inverses and applications (Proc. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973) Publ. Math. Res. Center Univ. Wisconsin, No. 32, Academic Press, New York, 1976, pp. 325–396. MR 0500249
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 145651, DOI 10.1090/S0002-9947-1962-0145651-8
W. V. Petryshyn, On generalized inverses and on the uniform convergence of $(I-\beta K)^{n}$ with application to iterative methods, J. Math. Anal. Appl. 18 (1967), 417–439. MR 208381, DOI 10.1016/0022-247X(67)90036-4