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

Böhmer, Klaus; Locker, John

doi:10.1090/S0025-5718-1988-0942144-4

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

Wolf-Jürgen Beyn, Discrete Green’s functions and strong stability properties of the finite difference method, Applicable Anal. 14 (1982/83), no. 2, 73–98. MR 678496

K. Böhmer, Fehlerasymptotik von Diskretisierungsverfahren und ihre numerische Anwendung, Interner Bericht Nr. 77/2, Institut für Praktische Mathematik, Universität Karlsruhe, 1977.

Klaus Böhmer, Discrete Newton methods and iterated defect corrections, Numer. Math. 37 (1981), no. 2, 167–192. MR 623039, DOI https://doi.org/10.1007/BF01398251
K. Böhmer and J. Locker, Asymptotic expansions in ill-posed boundary value problems, Z. Angew. Math. Mech. 61 (1981), no. 5, T272–T273. MR 648245

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.

J. Albrecht and L. Collatz (eds.), Numerische Behandlung von Differentialgleichungen. Band 3, Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], vol. 56, Birkhäuser Verlag, Basel, 1981 (German). MR 784038
Henning Esser, Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen, Numer. Math. 28 (1977), no. 1, 69–100 (German, with English summary). MR 461926, DOI https://doi.org/10.1007/BF01403858
Rolf Dieter Grigorieff, Zur Theorie linearer approximationsregulärer Operatoren. I, II, Math. Nachr. 55 (1973), 233–249; ibid. 55 (1973), 251–263 (German). MR 348533, DOI https://doi.org/10.1002/mana.19730550113
R. Kannan and John Locker, Continuous dependence of least squares solutions of linear boundary value problems, Proc. Amer. Math. Soc. 59 (1976), no. 1, 107–110. MR 409947, DOI https://doi.org/10.1090/S0002-9939-1976-0409947-X
H. B. Keller and V. Pereyra, Difference methods and deferred corrections for ordinary boundary value problems, SIAM J. Numer. Anal. 16 (1979), no. 2, 241–259. MR 526487, DOI https://doi.org/10.1137/0716018
Heinz-Otto Kreiss, Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972), 605–624. MR 373296, DOI https://doi.org/10.1090/S0025-5718-1972-0373296-3
John Locker and P. M. Prenter, Regularization with differential operators. I. General theory, J. Math. Anal. Appl. 74 (1980), no. 2, 504–529. MR 572669, DOI https://doi.org/10.1016/0022-247X%2880%2990145-6
John Locker and P. M. Prenter, Regularization with differential operators. I. General theory, J. Math. Anal. Appl. 74 (1980), no. 2, 504–529. MR 572669, DOI https://doi.org/10.1016/0022-247X%2880%2990145-6
John Locker and P. M. Prenter, Regularization and linear boundary value problems, Applicable Anal. 20 (1985), no. 1-2, 129–149. MR 808065, DOI https://doi.org/10.1080/00036818508839565
Frank Natterer, Regularisierung schlecht gestellter Probleme durch Projektionsverfahren, Numer. Math. 28 (1977), no. 3, 329–341 (German, with English summary). MR 488721, DOI https://doi.org/10.1007/BF01389972
Victor Pereyra, On improving an approximate solution of a functional equation by deferred corrections, Numer. Math. 8 (1966), 376–391. MR 203967, DOI https://doi.org/10.1007/BF02162981
Victor Pereyra, Iterated deferred corrections for nonlinear operator equations, Numer. Math. 10 (1967), 316–323. MR 221760, DOI https://doi.org/10.1007/BF02162030
David L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 84–97. MR 134481, DOI https://doi.org/10.1145/321105.321114

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.

Robert D. Russell, A comparison of collocation and finite differences for two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 1, 19–39. MR 451745, DOI https://doi.org/10.1137/0714003
Robert D. Skeel, A theoretical framework for proving accuracy results for deferred corrections, SIAM J. Numer. Anal. 19 (1982), no. 1, 171–196. MR 646602, DOI https://doi.org/10.1137/0719009
Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer-Verlag, New York-Heidelberg, 1973. Springer Tracts in Natural Philosophy, Vol. 23. MR 0426438

A. N. Tikhonov, "Solution of incorrectly formulated problems and the regularization method," Soviet Math. Dokl., v. 4, 1963, pp. 1035-1038.

A. N. Tikhonov, "Regularization of incorrectly posed problems," Soviet Math. Dokl., v. 4, 1963, pp. 1624-1627.

Gennadi Vainikko, Funktionalanalysis der Diskretisierungsmethoden, B. G. Teubner Verlag, Leipzig, 1976 (German). Mit Englischen und Russischen Zusammenfassungen; Teubner-Texte zur Mathematik. MR 0468159