A dynamical tikhonov regularization method for solving nonlinear ill-posed problems

CMES - Computer Modeling in Engineering and Sciences

Volumn 76, Issue 2, 2011, Pages 109-132

  Liu, Chein Shan ^a   Kuo, Chung Lun ^b  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

^b NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

Author keywords

Dynamical Tikhonov regularization; Ill posed systems; Landweber Scherzer iterative algorithm; Optimal vector driven algorithm (OVDA); Tikhonov regularization

Indexed keywords

DRIVEN SYSTEM; ILL-POSED SYSTEMS; ITERATIVE ALGORITHM; ITERATIVE SOLUTIONS; LANDWEBER-SCHERZER ITERATIVE ALGORITHM; NONLINEAR ILL-POSED PROBLEMS; NONLINEAR ORDINARY DIFFERENTIAL EQUATION; NUMERICAL EXAMPLE; OPTIMAL REGULARIZATION; OPTIMAL VECTOR DRIVEN ALGORITHM (OVDA); REGULARIZATION PARAMETERS; TIKHONOV; TIKHONOV METHOD; TIKHONOV REGULARIZATION; TIKHONOV REGULARIZATION METHOD;

ALGORITHMS; COMPUTATIONAL EFFICIENCY; EULER EQUATIONS; ITERATIVE METHODS; OPTIMIZATION; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETERIZATION; PROBLEM SOLVING;

NONLINEAR EQUATIONS;

EID: 80955148346     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (35)

1. Davidenko, D. (1953): On a new method of numerically integrating a system of nonlinear equations. Doklady Akad. Nauk SSSR, vol. 88, pp. 601-604.
2. Engl, H. W. (1987): Discrepancy principles for Tikhonov regularization of illposed problems leading to optimal convergence rates. J. Optim. Theory. Appl., vol. 52, pp. 209-215.
3. Farcas A.; Elliott L.; Ingham D. B.; Lesnic D. (2004): An inverse dual reciprocity method for hydraulic conductivity identification in steady ground flow. Adv. Water Resou., vol. 27, pp. 223-235. (Pubitemid 38473858)
4. Fan, C.M.; Liu, C.-S.; Yeih, Y. C. and Chan, H. F. (2010): The Scalar Homotopy Method for Solving Non-Linear Obstacle Problem CMC: Computers, Materials, & Continua, Vol. 15, No. 1, pp. 67-86.
5. Gfrerer, H. (1987): An a-posteriori parameter choices for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comp., vol. 49, pp. 507-522.
6. Hanke, M.; Neubauer, A.; Scherzer, O. (1995): A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math., vol. 72, pp. 21-37.
7. Hansen, P. C. (1992): Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev., vol. 34, pp. 561-580.
8. Hansen, P. C.; O'Leary, D. P. (1993): The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput., vol. 14, pp. 1487-1503.
9. Jin, Q. (2001): A discrete scheme of Landweber iteration for solving nonlinear ill-posed problems. J. Math. Anal. Appl., vol. 253, pp. 187-203. (Pubitemid 33380882)
10. Ku, C.-Y.; Yeih, W.; Liu, C.-S. (2010): Solving non-linear algebraic equations by a scalar Newton-homotopy continuation method. Int. J. Nonlinear Sci. Numer. Simul., vol. 11, pp. 435-450.
11. Kunisch, K.; Zou, J. (1998): Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, vol. 14, pp. 1247-1264. (Pubitemid 128557070)
12. Landweber, L. (1951): An iteration formula for Fredholm integral equations of the first kind. Amer. J. Math., vol. 73, pp. 615-624.
13. Li, L.; Han, B.; Wang, W. (2007): R-K type Landweber method for nonlinear ill-posed problems. J. Comp. Appl. Math., vol. 206, pp. 341-357. (Pubitemid 46817763)
14. Liu, C.-S. (2000a): A Jordan algebra and dynamic system with associator as vector field. Int. J. Non-Linear Mech., vol. 35, pp. 421-429.
15. Liu, C.-S. (2000b): Intermittent transition to quasiperiodicity demonstrated via a circular differential equation. Int. J. Non-Linear Mech., vol. 35, pp. 931-946.
16. Liu, C.-S. (2007): A study of type I intermittency of a circular differential equation under a discontinuous right-hand side. J. Math. Anal. Appl., vol. 331, pp. 547-566.
17. Liu, C.-S. (2010a): The Fictitious Time Integration Method to Solve the Spaceand Time-Fractional Burgers Equations CMC: Computers, Materials, & Continua, Vol. 15, No. 3, pp. 221-240.
18. Liu, C.-S. (2010b): A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation CMC: Computers, Materials, & Continua, Vol. 18, No. 1, pp. 1-20.
19. Liu, C.-S.; Atluri, S. N. (2009): A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems. CMES: Computer Modeling in Engineering and Science, vol. 43, pp. 253-276.
20. Liu, C.-S.; Atluri, S. N. (2011a): Simple "residual-norm" based algorithms, for the solution of a large system of non-linear algebraic equations, which converge faster than the Newton's method. CMES: Computer Modeling in Engineering and Science, vol. 71, pp. 279-304.
21. Liu, C.-S.; Atluri, S. N. (2011b): A method for solving a system of nonlinear algebraic equations, F(x)=0, using the system of ODEs with an optimum a in ?x= l[aF+(1a)BTF]; Bi j = ¶Fi=¶ xj. CMES: Computer Modeling in Engineering and Science, vol. 73, pp. 395-431.
22. Liu, C.-S.; Yeih,W.; Kuo, C.-L.; Atluri, S. N. (2009): A scalar homotopy method for solving an over/under-determined system of non-linear algebraic equations. CMES: Computer Modeling in Engineering and Science, vol. 53, pp. 47-71.
23. Lukas, M. A. (1998): Comparison of parameter choice methods for regularization with discrete noisy data. Inverse Problems, vol. 14, pp. 161-184. (Pubitemid 128555639)
24. Morozov, V. A. (1966): On regularization of ill-posed problems and selection of regularization parameter. J. Comp. Math. Phys., vol. 6, pp. 170-175.
25. Morozov, V. A. (1984): Methods for Solving Incorrectly Posed Problems. Springer, New York.
26. Neubauer, A. (2000): On Landweber iteration for nonlinear ill-posed problem in Hilbert scales. Numer. Math., vol. 85, pp. 309-328.
27. Polyanin, A. D.; Manzhirov, A. V. (2008): Handbook of Integral Equations. 2nd edition, Chapman & Hall/CRC, Taylor & Francis Group.
28. Ramlau, R. A. (1998): A modified Landweber method for inverse problems. Numer. Funct. Anal. Optim., vol. 20, pp. 79-98.
29. Resmerita, E. (2005): Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Problems, vol. 21, pp. 1303-1314. (Pubitemid 41090541)
30. Scherzer, O. (1995): Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl., vol. 194, pp. 911-934.
31. Scherzer, O. (1998): A modified Landweber iteration for solving parameter estimation problems. Appl. Math. Optim., vol. 38, pp. 45-68. (Pubitemid 128621126)
32. Tikhonov, A. N.; Arsenin, V. Y. (1977): Solutions of Ill-Posed Problems. John- Wiley & Sons, New York.
33. Wang, W.; Han, B.; Li, L. (2008): A Runge-Kutta type modified Landweber method for nonlinear ill-posed operator equations. J. Comp. Appl. Math., vol. 212, pp. 457-468. (Pubitemid 350266226)
34. Wang, Y.; Xiao, T. (2001): Fast realization algorithms for determining regularization parameters in linear inverse problems. Inverse Problems, vol. 17, pp. 281-291. (Pubitemid 32419738)
35. Xie, J.; Zou, J. (2002): An improved model function method for choosing regularization parameters in linear inverse problems. Inverse Problems, vol. 18, pp. 631-643. (Pubitemid 34720662)