Regularization for thorder linear boundary value problems using thorder differential operators
Authors:
D. A. Kouba and John Locker
Journal:
Trans. Amer. Math. Soc. 293 (1986), 229255
MSC:
Primary 47E05; Secondary 34B05, 47A50
MathSciNet review:
814921
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Abstract: Let and denote real Hilbert spaces, and let be a closed denselydefined linear operator having closed range. Given an element , we determine least squares solutions of the linear equation by using the method of regularization. Let be a third Hilbert space, and let be a linear operator with . Under suitable conditions on and and for each , we show that there exists a unique element which minimizes the functional , and the converge to a least squares solution of as . We apply our results to the special case where is an thorder differential operator in , and we regularize using for an thorder differential operator in with . Using an approximating space of Hermite splines, we construct numerical solutions to by the method of continuous least squares and the method of discrete least squares.
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 [1]
 U. Ascher, Discrete least squares approximations for ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), 478496. MR 491701 (81e:65043)
 [2]
 R. Bellman, A note on an inequality of E. Schmidt, Bull. Amer. Math. Soc. 50 (1944), 734736. MR 0010732 (6:61g)
 [3]
 F. J. Beutler and W. L. Root, The operator pseudoinverse in control and systems identification, Generalized Inverses and Applications (M. Z. Nashed, ed.), Academic Press, New York, 1976, pp. 397494. MR 0490311 (58:9658)
 [4]
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 [5]
 , Functional analysis, nonlinear differential equations, and the alternative method, Nonlinear Functional Analysis and Differential Equations (L. Cesari, R. Kannan and J. D. Schur, eds.), Marcel Dekker, New York, 1976, pp. 1197. MR 0487630 (58:7249)
 [6]
 S. D. Conte and C. deBoor, Elementary numerical analysis, 2nd ed., McGrawHill, New York, 1972.
 [7]
 P. J. Davis, Interpolation and approximation, Dover, New York, 1975. MR 0380189 (52:1089)
 [8]
 C. deBoor and B. Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582606. MR 0373328 (51:9528)
 [9]
 J. Douglas, Jr. and T. Dupont, Galerkin approximations for the two point boundary value problem using continuous, piecewise polynomial spaces, Numer. Math. 22 (1974), 99109. MR 0362922 (50:15360)
 [10]
 J. K. Hale, Applications of alternative problems, Lecture Notes, Brown Univ., Providence, R.I., 1971.
 [11]
 E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966. MR 0201039 (34:924)
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 V. V. Ivanov, The theory of approximate methods and their application to the numerical solution of singular integral equations, Noordhoff, Leyden, 1976. MR 0405045 (53:8841)
 [13]
 R. Kannan and J. Locker, Continuous dependence of least squares solutions of linear boundary value problems, Proc. Amer. Math. Soc. 59 (1976), 107110. MR 0409947 (53:13699)
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 T. Kato, Perturbation theory for linear operators, 2nd ed., SpringerVerlag, Berlin, Heidelberg and New York, 1976. MR 0407617 (53:11389)
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 D. Kouba, Regularization for thorder linear boundary value problems using thorder differential operators, Doctoral Dissertation, Colorado State Univ., 1982.
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 J. Locker, The generalized Green's function for an thorder linear differential operator, Trans. Amer. Math. Soc. 288 (1977), 243268. MR 0481204 (58:1340)
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 , Functional analysis and twopoint differential operators, Lecture Notes, Colorado State Univ., Fort Collins, 1982.
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 J. Locker and P. M. Prenter, Optimal and error estimates for continuous and discrete least squares methods for boundary value problems, SIAM J. Numer. Anal. 15 (1978), 11511160. MR 512688 (80d:65097)
 [19]
 , Regularization with differential operators. I. General theory, J. Math. Anal. Appl. 74 (1980), 504529. MR 572669 (83j:65062a)
 [20]
 , Regularization with differential operators. II. Weak least squares finite element solutions to first kind integral equations, SIAM J. Numer. Anal. 17 (1980), 247267. MR 567272 (83j:65062b)
 [21]
 , Regularization and linear boundary value problems (to appear).
 [22]
 , Representors and superconvergence of least squares finite element approximates, Applicable Anal. (to appear).
 [23]
 M. Z. Nashed, Approximate regularized solutions to improperly posed linear integral and operator equations, Constructive and Computational Methods for Differential and Integral Equations (D. Colton and R. P. Gilbert, eds.), Lecture Notes in Math., vol. 430, SpringerVerlag, Berlin, Heidelberg and New York, 1974, pp. 289332. MR 0501883 (58:19118)
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 D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 8497. MR 0134481 (24:B534)
 [25]
 P. M. Prenter, Splines and variational methods, Wiley, New York, 1975. MR 0483270 (58:3287)
 [26]
 P. Sammon, The discrete least squares method, Math. Comp. 31 (1977), 6065. MR 0431699 (55:4694)
 [27]
 M. H. Schultz, Error bounds for polynomial spline interpolation, Math. Comp. 24 (1970), 507515. MR 0275025 (43:783)
 [28]
 A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl. 4 (1963), 10351038.
 [29]
 , Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963), 16241627.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608149216
PII:
S 00029947(1986)08149216
Article copyright:
© Copyright 1986 American Mathematical Society
