Abstract
Using recent results about Tikhonov regularization of nonlinear ill-posed problems, we show that nonlinear index-2 differential-algebraic equations can be solved in a stable way via Tikhonov regularization in various ways. Convergence rates for the regularized solutions are derived.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Adams, Sobolev Spaces; Academic Press, New York 1975
K. E. Brenan, L. R. Engquist, Backward differentiation approximations of nonlinear differential algebraic systems; Math. Comp. 51 (1988), pp. 659–676
H. W. Engl, C. W. Groetsch (eds.), Inverse and Ill-Posed Problems, Academic Press, Orlando 1987
H. W. Engl, K. Klinisch, A. Neubauer, Convergence rates for Tikhonov regularization of non-linear ill-posed problems; Inverse Problems 5 (1989), pp. 523–540
C. W. Gear, Maintaining solution invariants in the numerical solution of ODE’s, SIAM J. Sci. Comput. 7 (1986), pp. 734–743
C. W. Gear, G. K. Gupta, B. Leimkuhler, Automatic integration of Euler-Lagrange equations with constraints; J. Comp. Appl. Math. 12 and 13, (1985), pp. 77–90
E. Griepentrog, R. März, Differential-Algebraic Equations and their Numerical Treatment, Teubner-Texte zur Math. 88, Teubner, Leipzig 1986
C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston 1984
E. Hairer, CH. Lubich, M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York 1989
M. Hanke, Linear differential-algebraic equations in spaces of integrable functions; J. Differ. Equations 79 (1989), pp. 14–30
M.Hanke, R.März, A.Neubauer, On the regularization of linear differential-algebraic equations; in [3], pp. 523–540
M. Hanke, R. März, A. Neubauer, On the regularization of a class of nontransferable differential-algebraic equations; J. Differ. Equations 73 (1988), pp. 119–132
E. Kamke, Differentialgleichungen: Lösungsmethode und Lösungen I, 8.Auflage; Akad. Verlagsgesellschaft Geest and Portig, Leipzig 1967
P. Lötstedt, L. R. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints I, SIAM J.Sci.Stat. Comput. 7 (1986), pp. 720–733
R. März, Higher Index Differential-Algebraic Equations: Analysis and Numerical Treatment, Banach Center Publ. XIV, Warszawa, Polish Scientific Publishers 1989
V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York 1976
M. Z. Nashed (ed.), Generalized Inverses and Applications, Academic Press, New York 1976
A.Neubauer, When do Sobolev spaces form a Hilbert scale? Proc.Amer.Math.Soc. 103 (1988), pp. 557–562
A. Neubauer, Tikhonov regularization for nonlinear ill-posed problems: optimal convergence rates and finite dimensional approximation, Inverse Problems 5 (1989), pp. 541–557
T. I. Seidman, C. R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems 5 (1989), pp. 227–238
A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems, Wiley, New York 1977
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Engl, H.W., Hanke, M., Neubauer, A. (1990). Tikhonov Regularization of Nonlinear Differential-Algebraic Equations. In: Sabatier, P.C. (eds) Inverse Methods in Action. Inverse Problems and Theoretical Imaging. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75298-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-75298-8_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-75300-8
Online ISBN: 978-3-642-75298-8
eBook Packages: Springer Book Archive