Convergence of a generalized pulse-spectrum technique (GPST) for inverse problems of -D diffusion equations in space-time domain

Authors:
X. Y. Liu and Y. M. Chen

Journal:
Math. Comp. **51** (1988), 477-489

MSC:
Primary 65P05; Secondary 35R30

DOI:
https://doi.org/10.1090/S0025-5718-1988-0958636-8

MathSciNet review:
958636

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of convergence of a special form of the generalized pulse-spectrum technique (GPST) for solving inverse problems of one-dimensional diffusion equations in space-time 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.

**[1]**Y. M. Chen, "Generalized pulse-spectrum technique (GPST) for solving problems in parameter identification,"*Proc. U.S.-China Workshop on Advances in Computational Engineering Mechs.*, Dalian, China, Sept. 5-9, 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. 366-385.**[3]**Y. M. Chen and J. Q. Liu,*A numerical algorithm for remote sensing of thermal conductivity*, J. Comput. Phys.**43**(1981), no. 2, 315–326. MR**640361**, https://doi.org/10.1016/0021-9991(81)90125-X**[4]**Y. M. Chen and J. Q. Liu,*A numerical algorithm for solving inverse problems of two-dimensional wave equations*, J. Comput. Phys.**50**(1983), no. 2, 193–208. MR**707198**, https://doi.org/10.1016/0021-9991(83)90063-3**[5]**Y. M. Chen and J. Q. Liu,*An iterative numerical algorithm for solving multiparameter inverse problems of evolutional partial differential equations*, J. Comput. Phys.**53**(1984), no. 3, 429–442. MR**739109**, https://doi.org/10.1016/0021-9991(84)90069-X**[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. 125-135.**[7]**Y. M. Chen and G. Q. Xie,*An iterative method for simultaneous determination of bulk and shear moduli and density variations*, J. Comput. Phys.**62**(1986), no. 1, 143–163. MR**825895**, https://doi.org/10.1016/0021-9991(86)90104-X**[8]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[9]**L. A. Kantorovich & G. P. Akilov,*Functional Analysis in Normed Spaces*, Pergamon Press, Oxford, 1964.**[10]**J. Q. Liu and Y. M. Chen,*An iterative algorithm for solving inverse problems of two-dimensional diffusion equations*, SIAM J. Sci. Statist. Comput.**5**(1984), no. 2, 255–269. MR**740845**, https://doi.org/10.1137/0905018**[11]**X. Y. Liu and Y. M. Chen,*A generalized pulse-spectrum technique (GPST) for determining time-dependent coefficients of one-dimensional diffusion equations*, SIAM J. Sci. Statist. Comput.**8**(1987), no. 3, 436–445. MR**883779**, https://doi.org/10.1137/0908043**[12]**Victor Pereyra,*Iterative methods for solving nonlinear least squares problems*, SIAM J. Numer. Anal.**4**(1967), 27–36. MR**0216732**, https://doi.org/10.1137/0704003**[13]**Y. N. Tang & Y. M. Chen, "Application of GPST algorithm to history matching of single-phase simulator models,"*Advances in Computer Methods for Partial Differential Equations V*(R. Vichnevetsky and R. Stepleman, eds.), IMACS, 1984, pp. 433-439.**[14]**Andrey N. Tikhonov and Vasiliy Y. Arsenin,*Solutions of ill-posed problems*, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR**0455365****[15]**G. Q. Xie and Y. M. Chen,*A modified pulse-spectrum technique for solving inverse problems of two-dimensional elastic wave equation*, Appl. Numer. Math.**1**(1985), no. 3, 217–237. MR**792358**, https://doi.org/10.1016/0168-9274(85)90017-0**[16]**G. Q. Xie, Y. M. Chen & X. Y. Liu, "Convergence of a generalized pulse-spectrum technique (GPST) for inverse problems of 1-D evolutional partial differential equations,"*SIAM J. Numer. Anal.*(To appear.)

Retrieve articles in *Mathematics of Computation*
with MSC:
65P05,
35R30

Retrieve articles in all journals with MSC: 65P05, 35R30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0958636-8

Article copyright:
© Copyright 1988
American Mathematical Society