Convergence of a generalized pulsespectrum technique (GPST) for inverse problems of D diffusion equations in spacetime domain
Authors:
X. Y. Liu and Y. M. Chen
Journal:
Math. Comp. 51 (1988), 477489
MSC:
Primary 65P05; Secondary 35R30
MathSciNet review:
958636
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Abstract: The problem of convergence of a special form of the generalized pulsespectrum technique (GPST) for solving inverse problems of onedimensional diffusion equations in spacetime domain is considered. Under the assumptions that a Tikhonov regularized solution exists and the derivative operator of the regularized forward problem at the regularized solution is invertible, the iterative solutions of this special GPST converge to the Tikhonov regularized solution in C norm if the initial guess is close enough to the Tikhonov regularized solution and the rate of convergence is at least linear.
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G. Q. Xie, Y. M. Chen & X. Y. Liu, "Convergence of a generalized pulsespectrum technique (GPST) for inverse problems of 1D evolutional partial differential equations," SIAM J. Numer. Anal. (To appear.)
 [1]
 Y. M. Chen, "Generalized pulsespectrum technique (GPST) for solving problems in parameter identification," Proc. U.S.China Workshop on Advances in Computational Engineering Mechs., Dalian, China, Sept. 59, 1983.
 [2]
 Y. M. Chen & D. S. Tsien, "A numerical algorithm for remote sensing of density profiles of a simple ocean model by acoustic pulses," J. Comput. Phys., v. 25, 1977, pp. 366385.
 [3]
 Y. M. Chen & J. Q. Liu, "A numerical algorithm for remote sensing of thermal conductivity," J. Comput. Phys., v. 43, 1981, pp. 315326. MR 640361 (83c:65207)
 [4]
 Y. M. Chen & J. Q. Liu, "A numerical algorithm for solving inverse problems of twodimensional wave equation," J. Comput. Phys., v. 50, 1983, pp. 193208. MR 707198 (84m:65114)
 [5]
 Y. M. Chen & J. Q. Liu, "An iterative numerical algorithm for solving multiparameter inverse problems of evolutional partial differential equations," J. Comput. Phys., v. 53, 1984, pp. 429442. MR 739109 (85i:65124)
 [6]
 Y. M. Chen & G. Q. Xie, "A numerical method for simultaneous determination of bulk modulus, shear modulus and density variations for nondestructive evaluation," Nondestructive Testing Communication, v. 1, 1984, pp. 125135.
 [7]
 Y. M. Chen & G. Q. Xie, "An iterative method for simultaneous determination of bulk and shear moduli and density variation," J. Comput. Phys., v. 62, 1986, pp. 143163. MR 825895 (87e:86008)
 [8]
 A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, Englewood Cliffs, N. J., 1964. MR 0181836 (31:6062)
 [9]
 L. A. Kantorovich & G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964.
 [10]
 J. Q. Liu & Y. M. Chen, "An iterative algorithm for solving inverse problems of twodimensional diffusion equations," SIAM J. Sci. Statist. Comput., v. 5, 1984, pp. 255269. MR 740845 (86c:65111)
 [11]
 X. Y. Liu & Y. M. Chen, "A generalized pulsespectrum technique for determining timedependent coefficients of onedimensional diffusion equations," SIAM J. Sci. Statist. Comput., v. 8, 1987, pp. 436445. MR 883779 (88k:35215)
 [12]
 V. Pereyra, "Iterative methods for solving nonlinear least squares problems," SIAM J. Numer. Anal., v. 4, 1967, pp. 2736. MR 0216732 (35:7561)
 [13]
 Y. N. Tang & Y. M. Chen, "Application of GPST algorithm to history matching of singlephase simulator models," Advances in Computer Methods for Partial Differential Equations V (R. Vichnevetsky and R. Stepleman, eds.), IMACS, 1984, pp. 433439.
 [14]
 A. N. Tikhonov & V. Y. Arsenin, Solutions of IllPosed Problems, Wiley, New York, 1977. MR 0455365 (56:13604)
 [15]
 G. Q. Xie & Y. M. Chen, "A modified pulsespectrum technique for solving inverse problems of twodimensional elastic wave equation," Appl. Numer. Math., v. 1, 1985, pp. 217237. MR 792358 (86e:73023)
 [16]
 G. Q. Xie, Y. M. Chen & X. Y. Liu, "Convergence of a generalized pulsespectrum technique (GPST) for inverse problems of 1D evolutional partial differential equations," SIAM J. Numer. Anal. (To appear.)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809586368
PII:
S 00255718(1988)09586368
Article copyright:
© Copyright 1988
American Mathematical Society
