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
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by Klaus Böhmer and John Locker PDF

Math. Comp. 51 (1988), 75-91 Request permission

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.

References

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

Fehlerasymptotik von Diskretisierungsverfahren und ihre numerische Anwendung

Klaus Böhmer, Discrete Newton methods and iterated defect corrections, Numer. Math. 37 (1981), no. 2, 167–192. MR 623039, DOI 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

The Mathematical Foundations of Asymptotic Expansions for the Discretization Error of Least Squares Solutions of Linear Boundary Value Problems

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, DOI 10.1007/978-3-0348-5454-2
Henning Esser, Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen, Numer. Math. 28 (1977), no. 1, 69–100 (German, with English summary). MR 461926, DOI 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 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 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 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 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 10.1016/0022-247X(80)90145-6
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 10.1016/0022-247X(80)90145-6
John Locker and P. M. Prenter, Regularization and linear boundary value problems, Applicable Anal. 20 (1985), no. 1-2, 129–149. MR 808065, DOI 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 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 10.1007/BF02162981
Victor Pereyra, Iterated deferred corrections for nonlinear operator equations, Numer. Math. 10 (1967), 316–323. MR 221760, DOI 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 10.1145/321105.321114

Philos. Trans. Roy. Soc. London Ser. A

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 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 10.1137/0719009
Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer Tracts in Natural Philosophy, Vol. 23, Springer-Verlag, New York-Heidelberg, 1973. MR 0426438, DOI 10.1007/978-3-642-65471-8

Soviet Math. Dokl.

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