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Asymptotic expansions for the discretization error of least squares solutions of linear boundary value problems


Authors: Klaus Böhmer and John Locker
Journal: Math. Comp. 51 (1988), 75-91
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1988-0942144-4
MathSciNet review: 942144
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Abstract: For determining least squares solutions of linear boundary value problems, the method of regularization provides uniquely solvable boundary value problems, which are solved with difference methods. The determination of the coefficients in an asymptotic expansion of the discretization error in powers of the regularization and discretization parameters $ \alpha $ and h, respectively, is an ill-posed problem. We present here an asymptotic expansion of this type and discuss the numerical implications for Richardson extrapolation, thereby establishing for the first time methods of arbitrarily high order.


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  • [1] W.-J. Beyn, "Discrete Green's functions and strong stability properties of the finite difference method," Applicable Anal., v. 14, 1982, pp. 73-98. MR 678496 (83m:65061)
  • [2] K. Böhmer, Fehlerasymptotik von Diskretisierungsverfahren und ihre numerische Anwendung, Interner Bericht Nr. 77/2, Institut für Praktische Mathematik, Universität Karlsruhe, 1977.
  • [3] K. Böhmer, "Discrete Newton methods and iterated defect corrections," Numer. Math., v. 37, 1981, pp. 167-192. MR 623039 (82j:65035)
  • [4] K. Böhmer & J. Locker, "Asymptotic expansions in ill-posed boundary value problems," Z. Angew. Math. Mech., v. 61, 1981, pp. T272-T273. MR 648245 (84c:65101)
  • [5] K. Böhmer & J. Locker, The Mathematical Foundations of Asymptotic Expansions for the Discretization Error of Least Squares Solutions of Linear Boundary Value Problems, Technical Report, Colorado State University, Fort Collins, Colorado, 1984.
  • [6] L. Collatz, The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin and New York, 1966. MR 784038 (86b:65003)
  • [7] H. Esser, "Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen," Numer. Math., v. 28, 1977, pp. 69-100. MR 0461926 (57:1908)
  • [8] R. D. Grigorieff, "Zur Theorie linearer approximations regulärer Operatoren, I und II," Math. Nachr., v. 55, 1973, pp. 233-249 and 251-263. MR 0348533 (50:1031)
  • [9] R. Kannan & J. Locker, "Continuous dependence of least squares solutions of linear boundary value problems," Proc. Amer. Math. Soc., v. 59, 1976, pp. 107-110. MR 0409947 (53:13699)
  • [10] H. B. Keller & V. Pereyra, "Difference methods and deferred corrections for ordinary boundary value problems," SIAM J. Numer. Anal., v. 16, 1979, pp. 241-259. MR 526487 (80h:65058)
  • [11] H. O. Kreiss, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605-624. MR 0373296 (51:9496)
  • [12] J. Locker & P. M. Prenter, "Regularization with differential operators. I. General theory," J. Math. Anal. Appl., v. 74, 1980, pp. 504-529. MR 572669 (83j:65062a)
  • [13] J. Locker & P. M. Prenter, "Regularization with differential operators. II: Weak least squares finite element solutions to first kind integral equations," SIAM J. Numer. Anal., v. 17, 1980, pp. 247-267. MR 567272 (83j:65062b)
  • [14] J. Locker & P. M. Prenter, "Regularization and linear boundary value problems," Applicable Anal., v. 20, 1985, pp. 129-149. MR 808065 (87f:47072)
  • [15] F. Natterer, "Regularisierung schlecht gestellter Probleme durch Projektionsverfahren," Numer. Math., v. 28, 1977, pp. 329-341. MR 0488721 (58:8238)
  • [16] V. Pereyra, "On improving an approximate solution of a functional equation by deferred corrections," Numer. Math., v. 8, 1966, pp. 376-391. MR 0203967 (34:3814)
  • [17] V. Pereyra, "Iterated deferred corrections for nonlinear operator equations," Numer. Math., v. 10, 1967, pp. 316-323. MR 0221760 (36:4812)
  • [18] D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach., v. 9, 1962, pp. 84-97. MR 0134481 (24:B534)
  • [19] L. F. Richardson, "The deferred approach to the limit. I: Single lattice," Philos. Trans. Roy. Soc. London Ser. A, v. 226, 1927, pp. 299-349.
  • [20] R. D. Russell, "A comparison of collocation and finite differences for two-point boundary value problems," SIAM J. Numer. Anal., v. 14, 1977, pp. 19-39. MR 0451745 (56:10027)
  • [21] R. D. Skeel, "A theoretical framework for proving accuracy results for deferred corrections," SIAM J. Numer. Anal., v. 19, 1982, pp. 171-196. MR 646602 (83d:65184)
  • [22] H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin and New York, 1973. MR 0426438 (54:14381)
  • [23] A. N. Tikhonov, "Solution of incorrectly formulated problems and the regularization method," Soviet Math. Dokl., v. 4, 1963, pp. 1035-1038.
  • [24] A. N. Tikhonov, "Regularization of incorrectly posed problems," Soviet Math. Dokl., v. 4, 1963, pp. 1624-1627.
  • [25] G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner--Texte zur Mathematik, Teubner, Leipzig, 1976. MR 0468159 (57:7997)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0942144-4
Keywords: Least squares solutions, regularization, ill-posed problems, asymptotic expansions for discretization, Richardson extrapolation
Article copyright: © Copyright 1988 American Mathematical Society

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