A numerical method for parameter identification of a boundary value problem

Applicable Analysis

Volumn 69, Issue 3-4, 1998, Pages 349-379

  Lin, Tao ^a   Ramirez, Edgardo ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Error estimates; Finite element; Inverse problem; Multigrid; Parameter identification; Regularization

Indexed keywords

EID: 2942513180     PISSN: 00036811     EISSN: 1563504X     Source Type: Journal    
DOI: 10.1080/00036819808840667     Document Type: Article

References (32)

1. Alessandrini, G., 1986. An identification problem for an elliptic equation in two variables. Ann. Mat. Pura. Appl., 145: 265–296.
2. Allgower, E.L., Böhmer, K., Potra, F.A., and Rheinboldt, W.C., 1986. A mesh-independence principle for operator equations and their discretizations. SIAM J. Numer. Anal., 23 (1): 160–169.
3. Allgower, E.L., and McCormick, S.F., 1978. Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems. Numer. Math., 29 (1): 237–260.
4. Banks, H.T., and Kunisch, K., 1989. Estimation Techniques for Distributed Parameter Systems, Boston: Birkhäuser.
5. Bazaraa, M.S., and Shetty, C.M., 1979. Nonlinear Programming, Theory and Algorithms, New York: John Wiley & Sons.
6. Bocheve, P.B., and Gunzburger, M.D., 1993. Accuracy of the least-squares methods for the navier-stokes equations. Comput. Fluids, 22: 545–563.
7. Cai, Z., Lazarov, R., Manfeuffel, T., and McCormick, S., 1994. First-order system least squares for second-order partial differential equations. Part I, SIAM Num. Anal., 31: 1785–1802.
8. Cai, Z., Manfeuffel, T., and McCormick, S., 1997. First-order system least squares for second-order partial differential equations. Part II, SIAM Num. Anal., 34: 425–454.
9. Carey, G.F., Pehlivanov, A., and Shen, Y., 1994. “ Least-squares mixed finite elements ”. In Finite Element Methods, Edited by: Kȓí[zbreve]ek, M., Neittaamäki, R., and Stenberg, R., 105–117. Marcel Dekker, Inc.
10. Chang, C.L., and Nelson, J.J., 1997. Least-squares finite element method for the stokes problem with zero residual of mass conservation. SIAM Num. Anal., 34: 480–489.
11. Chavent, G., 1980. “ Indentification of distributed parameter system: About the least squares method ”. In Proc. 5th IFAC Symposium on Identifiability and System Parameter Estimation, Edited by: Iserman, R., 85–97. Pergamon Press.
12. Dixon, L. C. W., Spedicato, E., and Szego, G. P., eds. 1980. Nonlinear optimisation, theory and algorithms, Birkhauser. The proceedings of an international summer school held at University of Bergamo, Italy, in September 1979 and repeated at the Hatfield Polytechnic, England, in July 1980).
13. Ewing, R., Lin, T., and Falk, R., 1987. “ Inverse and ill-posed problems in reservoir simulation ”. In Inverse and Ill-Posed Problems, Edited by: Engl, H.W., and Groetsch, C.W., 483–497. Academic Press, Inc.
14. Falk, R.S., 1983. Error estimates for the numerical identification of a variable coefficient. Math. Comp., 40: 537–546.
15. Frind, E.O., and Pinder, G.F., 1973. Galerkin solution of the inverse problem for aquifer transmissivity. Water Resources Research, 9: 1397–1410.
16. Ito, K., Kroller, M., and Kunisch, K., 1991. A numerical study of an augmented lagrangian method for the estimation of parameters in elliptic systems. SIAM J. Sci. and Stat. Computing, 12: 884–910.
17. Ito, K., and Kunisch, K., 1990. The augmented lagrangian method for parameter estimation in elliptic systems. SIAM J. Control and Optimation, 28: 113–136.
18. Ito, K., and Kunisch, K., 1990. The augmented lagrangian method for equality and inequality constrained problems in hilbert spaces. Mathematical Programming, 46: 341–460.
19. Kärkkäinen, T., 1996. Error estimates for distributed parameter identification in linear elliptic equations. Journal of Mathematical Systems, Estimation, and Control, 6 (1): 117–120.
20. Kohn, R.V., and Lowe, B.D., 1988. A variational method for parameter identification. RAIRO, Math. Model. and Numer. Anal, 22 (1): 119–158.
21. Kunisch, K., 1988. Inherent identifiability of parameters in elliptic differential equations. J. of Mathematical Analysis and Applications, 132 (1): 453–472.
22. Kunisch, K., and White, L., 1987. Identifiability under approximation for an elliptic boundary value problem. SIAM J. Control Optim., 25 (1): 279–297.
23. Lin T., A c° finite element method for an inverse problem. Applied Mathematics and Computation 1998 to appear
24. McCormick, G.P., 1983. “ Nonlinear Programming ”. In Theory, Algorithms, and Application, New York: John Wiley & Sons.
25. Polis, M.P., and Goodson, R.E., 1976. Parameter identification in distributed systems: a synthesizing overview. Proc. of the IEEE, 64: 45–61.
26. Richter, G.R., 1981. An inverse problem for the steady state diffusion equation. SIAM J. Appl. Math, 41: 210–221.
27. Richter, G.R., 1981. Numerical identification of a spatially varying diffusion coefficient. Math. Comp., 36: 375–386.
28. Rannacher, R., 1991. “ On the convergence of the newton-raphson method for strongly nonlinear ”. In Nonlinear Computational Mechanics, Edited by: Wriggers, P., 11–30. New York: Springer-Verlag. State of the Art
29. Schultz, M.H., 1973. Spline Analysis, Englewood Cliffs, N.J.: Prentice-Hall, Inc.
30. Yeh, W., 1986. Review of parameter identification procedures in groundwater hydrology: the inverse problem. Water Resource Research, 22: 95–108.
31. Zangwill, W.I., 1969. Nonlinear Programming: A Unified Approach, Englewood Cliffs, N.J.: Prentice-Hall.
32. Zeidler, E., 1990. Nonlinear Functional Analysis and its Application, II/B, New York: Springer-Verlag.