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

MathSciNet review:
0494922

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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]**Philip M. Anselone,*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR**0443383****[2]**P. M. Anselone and R. H. Moore,*Approximate solutions of integral and operator equations*, J. Math. Anal. Appl.**9**(1964), 268–277. MR**0184448****[3]**Kendall E. Atkinson,*The solution of non-unique linear integral equations*, Numer. Math.**10**(1967), 117–124. MR**0220013****[4]**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****[5]**Adi Ben-Israel,*On direct sum decompositions of Hestenes algebras*, Israel J. Math.**2**(1964), 50–54. MR**0171172****[6]**Adi Ben-Israel,*On error bounds for generalized inverses*, SIAM J. Numer. Anal.**3**(1966), 585–592. MR**0215504****[7]**A. BEN-ISRAEL, "A note on an iterative method for generalized inversion of matrices,"*Math. Comp.*, v. 20, 1966, pp. 439-440.**[8]**Adi Ben-Israel and Dan Cohen,*On iterative computation of generalized inverses and associated projections*, SIAM J. Numer. Anal.**3**(1966), 410–419. MR**0203917****[9]**Adi Ben-Israel and Thomas N. E. Greville,*Generalized inverses: theory and applications*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0396607****[10]**Abraham Berman and Robert J. Plemmons,*Cones and iterative methods for best least squares solutions of linear systems*, SIAM J. Numer. Anal.**11**(1974), 145–154. MR**0348984****[11]**J. Blatter, P. D. Morris, and D. E. Wulbert,*Continuity of the set-valued metric projection*, Math. Ann.**178**(1968), 12–24. MR**0228984****[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 and M. Z. Nashed,*On the convergence of the conjugate gradient method for singular linear operator equations*, SIAM J. Numer. Anal.**9**(1972), 165–181. MR**0319368****[15]**W. J. Kammerer and R. J. Plemmons,*Direct iterative methods for least-squares solutions to singular operator equations*, J. Math. Anal. Appl.**49**(1975), 512–526. MR**0368418****[16]**L. V. KANTOROVICH, "Functional analysis and applied mathematics,"*Uspehi Mat. Nauk*, v. 3, 1948, pp. 89-195. (Russian)**[17]**L. V. Kantorovich and G. P. Akilov,*Functional analysis in normed spaces*, Translated from the Russian by D. E. Brown. Edited by A. P. Robertson. International Series of Monographs in Pure and Applied Mathematics, Vol. 46, The Macmillan Co., New York, 1964. MR**0213845****[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.**9**(1978), no. 6, 1083–1095. MR**512512**, 10.1137/0509087**[20]**V. Lovass-Nagy and D. L. Powers,*On under- and over-determined initial value problems*, Internat. J. Control**19**(1974), 653–656. MR**0350094****[21]**L. A. Liusternik and V. J. Sobolev,*Elements of functional analysis*, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. 5, Hindustan Publishing Corp., Delhi; Gordon and Breach Publishers, Inc., New York, 1961. MR**0141967****[22]**R. H. Moore and M. Z. Nashed,*Approximations to generalized inverses of linear operators*, SIAM J. Appl. Math.**27**(1974), 1–16. MR**0361858****[23]**F. J. Murray,*On complementary manifolds and projections in spaces 𝐿_{𝑝} and 𝑙_{𝑝}*, Trans. Amer. Math. Soc.**41**(1937), no. 1, 138–152. MR**1501894**, 10.1090/S0002-9947-1937-1501894-5**[24]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[25]**M. Z. Nashed,*Generalized inverses, normal solvability, and iteration for singular operator equations*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 311–359. MR**0275246****[26]**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) Academic Press, New York, 1976, pp. 325–396. Publ. Math. Res. Center Univ. Wisconsin, No. 32. MR**0500249****[27]**T. G. Newman and P. L. Odell,*On the concept of a 𝑝-𝑞 generalized inverse of a matrix*, SIAM J. Appl. Math.**17**(1969), 520–525. MR**0255559****[28]**W. V. Petryshyn,*On generalized inverses and on the uniform convergence of (𝐼-𝛽𝐾)ⁿ with application to iterative methods*, J. Math. Anal. Appl.**18**(1967), 417–439. MR**0208381****[29]**W. V. Petryshyn,*On the generalized overrelaxation method for operation equations*, Proc. Amer. Math. Soc.**14**(1963), 917–924. MR**0169402**, 10.1090/S0002-9939-1963-0169402-2**[30]**W. V. Petryshyn,*On the extrapolated Jacobi or simultaneous displacements method in the solution of matrix and operator equations*, Math. Comp.**19**(1965), 37–55. MR**0176601**, 10.1090/S0025-5718-1965-0176601-2**[31]**J. L. PHILLIPS,*Collocation as a Projection Method for Solving Integral and Other Operator Equations*, Thesis, Purdue University, 1969.**[32]**P. M. Prenter,*A collection method for the numerical solution of integral equations*, SIAM J. Numer. Anal.**10**(1973), 570–581. MR**0327064****[33]**C. Radhakrishna Rao and Sujit Kumar Mitra,*Generalized inverse of matrices and its applications*, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0338013****[34]**Ivan Singer,*Bases in Banach spaces. I*, Springer-Verlag, New York-Berlin, 1970. Die Grundlehren der mathematischen Wissenschaften, Band 154. MR**0298399****[35]**Ivar Stakgold,*Boundary value problems of mathematical physics. Vol. I*, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1967. MR**0205776****[36]**G. W. Stewart,*On the continuity of the generalized inverse*, SIAM J. Appl. Math.**17**(1969), 33–45. MR**0245583****[37]**Angus E. Taylor,*Introduction to functional analysis*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0098966****[38]**K. S. Thomas,*On the approximate solution of operator equations*, Numer. Math.**23**(1974/75), 231–239. MR**0373275****[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]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[41]**S. Zlobec,*On computing the best least squares solutions in Hilbert space*, Rend. Circ. Mat. Palermo (2)**25**(1976), no. 3, 256–270 (1977) (English, with Italian summary). MR**0519596**

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Additional Information

DOI:
http://dx.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