Dynamical systems gradient method for solving Ill-conditioned linear algebraic systems

Acta Applicandae Mathematicae

Volumn 111, Issue 2, 2010, Pages 189-204

  Hoang, N S ^a   Ramm, A G ^a  

^a KANSAS STATE UNIVERSITY   (United States)

Author keywords

Dynamical Systems Method (DSM); Ill conditioned linear algebraic systems; Variational Regularization

Indexed keywords

APRIORI; DYNAMICAL SYSTEMS METHOD; ILL-CONDITIONED; LINEAR ALGEBRAIC SYSTEMS; MATRIX; NUMERICAL RESULTS; POSTERIORI; SPECTRAL DECOMPOSITION; STOPPING RULE; VARIATIONAL REGULARIZATION;

COMPUTATION THEORY; DYNAMICAL SYSTEMS; GRADIENT METHODS; NUMERICAL ANALYSIS;

ALGEBRA;

EID: 77955575346     PISSN: 01678019     EISSN: 15729036     Source Type: Journal    
DOI: 10.1007/s10440-009-9540-3     Document Type: Article

References (24)

1. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equation. I. Nonstiff Problems. Springer, Berlin (1987)
2. N.S. Hoang A.G. Ramm 2008 Solving ill-conditioned linear algebraic systems by the dynamical systems method (DSM) Inverse Probl. Sci. Eng. 16 N5 617 630 10.1080/17415970701743335 2440844
3. N.S. Hoang A.G. Ramm 2008 On stable numerical differentiation Aust. J. Math. Anal. Appl. 5 N1 1 7 10.1155/2008/371632 2399303 Article 5
4. N.S. Hoang A.G. Ramm 2008 An iterative scheme for solving nonlinear equations with monotone operators BIT Numer. Math. 48 N4 725 741 1169.65047 10.1007/s10543-008-0199-3 2465700
5. N.S. Hoang A.G. Ramm 2009 Dynamical systems method for solving linear finite-rank operator equations Ann. Pol. Math. 95 N1 77 93 1161.47059 10.4064/ap95-1-6 2466015
6. N.S. Hoang A.G. Ramm 2009 Dynamical Systems Gradient method for solving nonlinear equations with monotone operators Acta Appl. Math. 106 473 499 1165.47049 10.1007/s10440-008-9308-1 2504731
7. N.S. Hoang A.G. Ramm 2009 A new version of the Dynamical Systems Method (DSM) for solving nonlinear equations with monotone operators Differ. Equ. Appl. 1 N1 1 25 1172.47044 2508969
8. N.S. Hoang A.G. Ramm 2009 A discrepancy principle for equations with monotone continuous operators Nonlinear Anal., Theory Methods Appl. 70 4307 4315 05551973 10.1016/j.na.2008.09.014 2514761
9. N.S. Hoang A.G. Ramm 2010 Dynamical systems method for solving nonlinear equations with monotone operators Math. Comput. 79 269 2552225
10. Hoang, N.S., Ramm, A.G.: The Dynamical Systems Method for solving nonlinear equations with monotone operators. Asian Eur. Math. J. (2009, to appear)
11. Ivanov, V., Tanana, V., Vasin, V.: Theory of Ill-Posed Problems. VSP, Utrecht (2002)
12. Lattes, J., Lions, J.: Mèthode de Quasi- Réversibilité et Applications. Dunod, Paris (1967)
13. Morozov: Methods of Solving Incorrectly Posed Problems. Springer, New York (1984)
14. J.G. Nagy K.M. Palmer 2003 Steepest descent, CG, and iterative regularization of ill-posed problems BIT Numer. Math. 43 1003 1017 1045.65034 10.1023/B:BITN.0000014546.51341.53 2058881 (Pubitemid 39064026)
15. A.G. Ramm 2004 Dynamical systems method for solving operator equations Commun. Nonlinear Sci. Numer. Simul. 9 N2 383 402 1073.47056 10.1016/S1007-5704(03)00006-6 2045643 (Pubitemid 38398540)
16. Ramm, A.G.: Dynamical Systems Method (DSM) and nonlinear problems. In: Lopez-Gomez, J. (ed.) Spectral Theory and Nonlinear Analysis, pp. 201-228. World Scientific, Singapore (2005)
17. A.G. Ramm 2006 Dynamical systems method (DSM) for unbounded operators Proc. Am. Math. Soc. 134 N4 1059 1063 1095.47022 10.1090/S0002-9939-05-08076-7 (Pubitemid 43555442)
18. A.G. Ramm 2006 Dynamical systems method for nonlinear equations in Banach spaces Commun. Nonlinear Sci. Numer. Simul. 11 N3 306 310 1083.65056 10.1016/j.cnsns.2004.12.004 2176512 (Pubitemid 41448049)
19. Ramm, A.G.: Dynamical Systems Method for Solving Operator Equations. Elsevier, Amsterdam (2007)
20. A.G. Ramm 2009 Dynamical systems method for solving linear ill-posed problems Ann. Pol. Math. 95 N3 253 272 05533909 10.4064/ap95-3-5
21. Ramm, A.G., Airapetyan, R.: Dynamical systems and discrete methods for solving nonlinear ill-posed problems. In: Anastassiou, G. (ed.) Appl. Math. Reviews, vol. 1, pp. 491-536. World Scientific, Singapore (2000)
22. A.G. Ramm A.B. Smirnova 2001 On stable numerical differentiation Math. Comput. 70 1131 1153 0973.65015 10.1090/S0025-5718-01-01307-2 1826578
23. A.G. Ramm A.B. Smirnova 2003 Stable numerical differentiation: when is it possible? J. Korean SIAM 7 N1 47 61 1969742
24. Vainikko, G., Veretennikov, A.: Iterative Processes in Ill-Posed Problems. Nauka, Moscow (1996)