Calculating the best approximate solution of an operator equation

Authors:
H. Wolkowicz and S. Zlobec

Journal:
Math. Comp. **32** (1978), 1183-1213

MSC:
Primary 65J05; Secondary 47A50

DOI:
https://doi.org/10.1090/S0025-5718-1978-0494922-X

MathSciNet review:
0494922

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques. Another approach is to extend the classical approximation theory of Kantorovich for equations with invertible operators to the singular case. The best approximate solution is now obtained as the limit of the best approximate solutions of simpler equations, usually systems of linear algebraic equations. In particular, a Galerkin-type method is formulated and its convergence to the best approximate solution is established. The methods of this paper can also be used for calculating the best least squares solution in Hilbert spaces or the true solution in the case of an invertible operator.

**[1]**P. M. ANSELONE,*Collectively Compact Approximation Theory*, Prentice-Hall, Englewood Cliffs, N.J., 1971. MR**0443383 (56:1753)****[2]**P. M. ANSELONE & R. H. MOORE, "Approximate solutions of integral and operator equations,"*J. Math. Anal. Appl.*, v. 9, 1964, pp. 268-277. MR**0184448 (32:1920)****[3]**K. E. ATKINSON, "The solution of non-unique linear integral equations,"*Numer. Math.*, v. 10, 1967, pp. 117-124. MR**0220013 (36:3082)****[4]**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)****[5]**A. BEN-ISRAEL, "On direct sum decompositions of Hestenes algebras,"*Israel J. Math.*, v. 2, 1964, pp. 50-54. MR**0171172 (30:1403)****[6]**A. BEN-ISRAEL, "On error bounds for generalized inverses,"*SIAM J. Numer. Anal.*, v. 3, 1966, pp. 585-592. MR**0215504 (35:6344)****[7]**A. BEN-ISRAEL, "A note on an iterative method for generalized inversion of matrices,"*Math. Comp.*, v. 20, 1966, pp. 439-440.**[8]**A. BEN-ISRAEL & DAN COHEN, "On iterative computation of generalized inverses and associated projections,"*J. SIAM Numer. Anal.*, v. 3, 1966, pp. 410-419. MR**0203917 (34:3764)****[9]**A. BEN-ISRAEL & T. N. E. GREVILLE,*Generalized Inverses, Theory and Applications*, Interscience, New York, 1974. MR**0396607 (53:469)****[10]**A. BERMAN & R. J. PLEMMONS, "Cones and iterative methods for best least squares solutions of linear systems,"*SIAM J. Numer. Anal.*, v. 11, 1974, pp. 145-154. MR**0348984 (50:1478)****[11]**J. BLATHER, P. D. MORRIS & D. E. WULBERT, "Continuity of set-valued metric projection,"*Math. Ann.*, v. 178, 1968, pp. 12-24. MR**0228984 (37:4563)****[12]**C. W. GROETSCH,*Computational Theory of Generalized Inverses of Bounded Linear Operators*:*Representation and Approximation*, Dekker, New York, 1977.**[13]**Y. IKEBE,*The Galerkin Method for the Numerical Solution of Fredholm Integral Equations of the Second Kind*, Rept. CNA-S, Univ. of Texas, Austin, Texas, 1970.**[14]**W. J. KAMMERER & M. Z. NASHED, "On the convergence of the conjugate gradient method for singular linear operator equations,"*SIAM J. Numer. Anal.*, v. 9, 1972, pp. 165-181. MR**0319368 (47:7912)****[15]**W. J. KAMMERER & R. J. PLEMMONS, "Direct iterative methods for least squares solutions to singular operator equations,"*J. Math. Anal. Appl.*, v. 49, 1975, pp. 512-526. MR**0368418 (51:4659)****[16]**L. V. KANTOROVICH, "Functional analysis and applied mathematics,"*Uspehi Mat. Nauk*, v. 3, 1948, pp. 89-195. (Russian)**[17]**L. V. KANTOROVICH & G. P. AKILOV,*Functional Analysis in Normed Spaces*, English transl., Pergamon Press, Oxford, 1964. MR**0213845 (35:4699)****[18]**M. A. KRASNOSEL'SKIÏ ET AL.,*Approximate Solution of Operator Equations*, Noordhoff, Groningen, 1972.**[19]**W. F. LANGFORD, "The generalized inverse of an unbounded linear operator with unbounded constraints,"*SIAM J. Math. Anal.*(To appear.) MR**512512 (81g:65070)****[20]**V. LOVASS-NAGY & D. L. POWERS, "On under- and over-determined initial value problems,"*Internat. J. Control*, v. 19, 1974, pp. 653-656. MR**0350094 (50:2587)****[21]**L. A. LIUSTERNIK & V. I. SOBOLEV,*Elements of Functional Analysis*, Ungar, New York, 1961. MR**0141967 (25:5362)****[22]**R. H. MOORE & M. Z. NASHED, "Approximations to generalized inverses of linear operators,"*SIAM J. Appl. Math.*, v. 27, 1974, pp. 1-16. MR**0361858 (50:14301)****[23]**F. J. MURRAY, "On complementary manifolds and projections in spaces and ,"*Trans. Amer. Math. Soc.*, v. 41, 1937, pp. 138-152. MR**1501894****[24]**N. I. MUSKHELISHVILI,*Singular Integral Equations*, Noordhoff, Groningen, 1946. MR**0355494 (50:7968)****[25]**M. Z. NASHED, "Generalized inverses, normal solvability, and iteration for singular operator equations," in*Nonlinear Functional Analysis and Applications*(L. B. RALL, editor), Academic Press, New York, 1971, pp. 311-359. MR**0275246 (43:1003)****[26]**M. Z. NASHED, "Perturbations and approximations for generalized inverses and linear operator equations," in*Generalized Inverses and Applications*(M. Z. NASHED, editor), Academic Press, New York, 1976, pp. 325-396. MR**0500249 (58:17923)****[27]**T. G. NEWMAN & P. L. ODELL, "On the concept of a*p - q*generalized inverse of a matrix,"*SIAM J. Appl. Math.*, v. 17, 1969, pp. 520-525. MR**0255559 (41:220)****[28]**W. V. PETRYSHYN, "On generalized inverses and on the uniform convergence of with application to iterative methods,"*J. Math. Anal. Appl.*, v. 18, 1967, pp. 417-439. MR**0208381 (34:8191)****[29]**W. V. PETRYSHYN, "On the generalized overrelaxation method for operator equations,"*Proc. Amer. Math. Soc.*, v. 14, 1963, pp. 917-924. MR**0169402 (29:6652)****[30]**W. V. PETRYSHYN, "On the extrapolated Jacobi or simultaneous displacements method in the solution of matrix and operator equations,*Math. Comp.*, v. 19, 1965, pp. 37-56. MR**0176601 (31:873)****[31]**J. L. PHILLIPS,*Collocation as a Projection Method for Solving Integral and Other Operator Equations*, Thesis, Purdue University, 1969.**[32]**P. M. PRENTER, "Collocation method for the numerical solution of integral equations,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 570-581. MR**0327064 (48:5406)****[33]**C. R. RAO & S. K. MITRA,*Generalized Inverse of Matrices and its Application*, Wiley, New York, 1971. MR**0338013 (49:2780)****[34]**I. SINGER,*Bases in Banach Spaces*, I, Springer-Verlag, New York, 1970. MR**0298399 (45:7451)****[35]**I. STAKGOLD,*Boundary Value Problems of Mathematical Physics*, Vol. I, Macmillan Series in Advanced Mathematica and Theoretical Physics, Macmillan, New York, 1969. MR**0205776 (34:5602)****[36]**G. W. STEWART, "On the continuity of the generalized inverse,"*SIAM J. Appl. Math.*, v. 17, 1969, pp. 33-45. MR**0245583 (39:6889)****[37]**A. E. TAYLOR,*Introduction to Functional Analysis*, Wiley, New York, 1958. MR**0098966 (20:5411)****[38]**K. S. THOMAS, "On the approximate solution of operator equations,"*Numer. Math.*, v. 23, 1975, pp. 231-239. MR**0373275 (51:9475)****[39]**R. S. VARGA,*Extensions of the Successive Overrelaxation Theory with Applications to Finite Element Approximations*, Department of Mathematics, Kent State University, Kent, Ohio.**[40]**R. S. VARGA,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR**0158502 (28:1725)****[41]**S. ZLOBEC, "On computing the best least squares solutions in Hubert spaces",*Rend. Circ. Mat. Palermo Ser.*II, v. 25, 1976, pp. 1-15. MR**0519596 (58:24929)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65J05,
47A50

Retrieve articles in all journals with MSC: 65J05, 47A50

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0494922-X

Keywords:
Best approximate solution,
inconsistent equation,
generalized inverse of an operator,
iterative methods,
Kantorovich's theory of approximation methods,
Galerkin's method

Article copyright:
© Copyright 1978
American Mathematical Society