Convergence of a generalized pulse-spectrum technique (GPST) for inverse problems of 1-D diffusion equations in space-time domain

Liu, X. Y.; Chen, Y. M.

doi:10.1090/S0025-5718-1988-0958636-8

Convergence of a generalized pulse-spectrum technique (GPST) for inverse problems of $1$-D diffusion equations in space-time domain
HTML articles powered by AMS MathViewer

by X. Y. Liu and Y. M. Chen PDF

Math. Comp. 51 (1988), 477-489 Request permission

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.

References

Proc. U.S.-China Workshop on Advances in Computational Engineering Mechs.

J. Comput. Phys.

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, DOI 10.1016/0021-9991(81)90125-X
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, DOI 10.1016/0021-9991(83)90063-3
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, DOI 10.1016/0021-9991(84)90069-X

Nondestructive Testing Communication

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, DOI 10.1016/0021-9991(86)90104-X
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836

Functional Analysis in Normed Spaces

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, DOI 10.1137/0905018
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, DOI 10.1137/0908043
Victor Pereyra, Iterative methods for solving nonlinear least squares problems, SIAM J. Numer. Anal. 4 (1967), 27–36. MR 216732, DOI 10.1137/0704003

Advances in Computer Methods for Partial Differential Equations V

Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, 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. MR 0455365
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, DOI 10.1016/0168-9274(85)90017-0

SIAM J. Numer. Anal.