Summary
For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alifanov, O.M., Rumjancev, S.V.: On the stability of iterative methods for the solution of linear ill-posed problems, Soviet Math. Dokl.20, 1133–1136 (1979)
Brakhage, H.: On ill-posed problems and the method of conjugate gradients. In: Engl, H.W., Groetsch, C.W. (eds.) Proc. of the Alpine-U.S. Seminar on Inverse and Ill-Posed Problems, pp. 165–175. Boston: Academic Press 1986
Daniel, J.W.: The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal.4, 10–26 (1967)
Engl, H.W.: Necessary and sufficient conditions for convergence of regularization methods for solving linear operator equations of the first kind, Numer. Funct. Anal. Optimization3, 201–222 (1981)
Gilyazov, S.F.: Iterative solution methods for inconsistent linear equations with nonselfadjoint operators, Moscow Univ. Comput. Math. & Cybern.1, 8–13 (1977)
Gilyazov, S.F., Regularizing algorithms based on the conjugate-gradient method, U.S.S.R. Comput. Maths. Math. Phys.26, 8–13 (1986)
Groetsch, C.W.: Generalized inverses of linear operators, 1st Ed. New York: Dekker 1977
Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind, 1st Ed. Boston: Pitman 1984
Krasnosel'skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutiskii, Ya.B., Stetsenko, V.Ya.: Approximate solution of operator equations (engl. translation from russian), 1st Ed. Groningen: Wolters-Noordhoff 1972
Langlois, W.E.: Conditions for termination of the method of steepest descent after a finite number of iterations. IBM J. Res. Develop.10, 98–99 (1966)
Louis, A.K.: Convergence of the conjugate gradient method for compact operators. In: Engl, H.W., Groetsch, C.W. (eds.) Proc of the Alpine-U.S. Seminar on Inverse and Ill-Posed Problems, pp. 177–183, 1st Ed. Boston: Academic Press 1986
Louis, A.K.: Inverse und schlecht gestellte Probleme,xx Ed. Stuttgart: Teubner 1989
McCormick, S.F., Rodrigue, G.H.: A uniform approach to gradient methods for linear operator equations. J. Math. Anal. Appl.49, 275–285 (1975)
Nemirov'skii, A.S.: The regularizing properties of the adjoint gradient method in ill-posed problems. U.S.S.R. Comput. Maths. Math. Phys.26, 7–16 (1986)
Nemirov'skii, A.S., Polyak, B.T.: Iterative methods for solving linear ill-posed problems under precise information I. Engineering Cybernetics22, 1–11 (1984)
Nemirov'skii, A.S., Polyak, B.T.: Iterative methods for solving linear ill-posed problems under precise information II, Engineering Cybernetics22, 50–56 (1984)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables, 1st Ed. New York: Academic Press 1970
Plato, R.: Optimal algorithms for linear ill-posed problems yield regularization methods. Numer. Funct. Anal. Optimization (accepted)
Samanskii, V.E.: Certain calculational schemes for iterative processes (russian). Ukrain. Math. Zh.14, 100–109 (1962)
van der Sluis, A., van der Vorst, H.A.: The rate of convergence of conjugate gradients. Numer. Math.48, 543–560 (1986)
Winther, R.: Some superlinear convergence results for the conjugate gradient method. SIAM J. Numer. Anal.17, 14–17 (1980)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eicke, B., Louis, A.K. & Plato, R. The instability of some gradient methods for ill-posed problems. Numer. Math. 58, 129–134 (1990). https://doi.org/10.1007/BF01385614
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01385614