Abstract
In this paper, adaptive finite element method is developed for the estimation of distributed parameter in elliptic equation. Both upper and lower error bound are derived and used to improve the accuracy by appropriate mesh refinement. An efficient preconditioned project gradient algorithm is employed to solve the nonlinear least-squares problem arising in the context of parameter identification problem. The efficiency of our error estimators is demonstrated by some numerical experiments.
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Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)
Ascher, U., Haber, E.: Grid refinement and scaling for distributed parameter identification problems. Inverse Problems 17, pp. 571–590 (2001)
Bangerth, W.: Adaptive finite elment methods for the identification of distributed parameters in the partial differential equations. PhD thesis, University of Heidelberg, Institut für Angewandte Mathematik (2002)
Bank, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter System. Birkhaäuser, Boston (1989)
Chan, T.F., Tai, X.C.: Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comput. Phys. 193, 40–66 (2003)
Chan, T.F., Tai, X.C.: Identification of discontinuous coefficients from elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25(3), 881–904 (2003)
Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84, 527–548 (2000)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Dobson, D.C., Santosa, F.: An image enhancement techneque for electrical impedance tomography. Inverse Problems 10, 317–334 (1994)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)
Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic Problems I: A linear model problem. SIAM J. Numer. Anal. 28, 43–77 (1991)
Ewing, R. (ed.) The Mathematics of Reservior Simulation. Frontiers Appl. Math. 1, SIAM, Philadelphia (1984)
Ewing, R.E., Lin, T., Falk, R.S.: Inverse and ill-posed problems in reservoir simulation. In: Inverse and Ill-Posed Problem, pp. 483–497. Academic, Boston, MA (1987)
Ewing, R.E., Lin, T.: Parameter identification problems in single-phase and two-phase flow. In: Proc. 4th Int. Conf. on Control of Distributed Parameter Systems, International Series for Numerical Mathematics 91, pp. 85–108. Birkhauser Verlag, Basel (1989)
Falk, R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973)
Falk, R.S.: Error estimates of the numerical identification of a variable coefficient. Math. Comp. 40, 537–546 (1983)
Feng, T., Gulliksson, M., Liu, W.B.: Adaptive finite element methods for identification of elastic constants. J. Sci. Comput. 26, 217–235 (2006)
Fletcher, R.: Practical Methods of Optimization. Wiley, New York (2000)
French, D.A., King, J.T.: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 12, 299–314 (1991)
Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numer. 13, 313–328 (1979)
Gunzburger, M.D., Hou, S.L.: Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J. Control Optim. 34, 1001–1043 (1996)
Haber, E., Ascher, U.M., Oldenburg, D.: On optimization techniques for solving nonlinear inverse problems. Inverse Problems 16, 1263–1280 (2000)
Hettlich, F., Rundell, W.: The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14(1), 67–82 (1998)
Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2002)
Huang, Y.Q., Li, R., Liu, W.B., Yan, N.N.: Efficient discretization for finite element approximation of constrained optimal control problems. (submitted)
Kärkkäinen, T.: Error estimates for distributed parameter identification problems. PhD thesis, University of Jyväskylä, Department of Mathematics (1995)
Knowles, G.: Finite dimensional approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20, 414–427 (1982)
Kufner, A., John, O., Fucik, S.: Function Spaces. Nordhoff, Leyden, The Netherlands (1977)
Kunisch, K., Liu, W.B., Yan, N.N.: A posteriori error estimates for a model parameter identification problem. In: EUNMA’01 Proceedings, pp. 723–730 (2002)
Ladyzhenskaya, O.A., Urlatseva, H.H.: Linear and Qusilinear Elliptic Equations. Academic, New York (1968)
Li, R., Lin, W.B., Ma, H.P., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002)
Li, R.: On multi-mesh h-adaptive algorithm. J. Sci. Comput. 24(3), 321–341 (2005)
Liu, W.B., Yan, N.N.: A posteriori error estimators for a class of variational inequalities. J. Sci. Comput. 35, 361–393 (2000)
Liu, W.B., Ma, H.P., Tang, T.: On mixed error estimates for elliptic obstacle problems. Adv. Comput. Math. 15, 261–283 (2001)
Lowe, B.D., Rundell, W.: Unique recovery of a coefficient in an elliptic equation from input sources. Inverse Problems 11(1), 211–215 (1995)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)
Sun, N.Z.: Inverse Problem in Groundwater Modeling. Kluwer, The Netherlands (1994)
Ulbrich, M.: Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. Habilitation thesis, Fakultät für Mathematik, Technische Universität München (2002)
Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, New York (1995)
Yeh, W.W.-G.: Review of parameter identification procedures in groundwater hydrology: the inverse problems. Water Resour. Rev. 22, 95–108 (1986)
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Communicated by A. Zhou.
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Feng, T., Yan, N. & Liu, W. Adaptive finite element methods for the identification of distributed parameters in elliptic equation. Adv Comput Math 29, 27–53 (2008). https://doi.org/10.1007/s10444-007-9035-6
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DOI: https://doi.org/10.1007/s10444-007-9035-6
Keywords
- Parameter identification
- Finite element approximation
- Adaptive finite element methods
- Least-squares
- Gauss–Newton