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Identification of nonlinear elliptic equations. (English) Zbl 0865.35139

The authors study the identification of the nonlinearities \(a\) and \(b\) in the following boundary value problems: \[ -\text{div}(a(\nabla y))\ni f\quad\text{in }\Omega,\quad y=0\quad\text{on }\partial\Omega \] and \[ -\Delta y+b(y)\ni f\quad\text{in }\Omega,\quad y=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with Lipschitzian boundary, and \(f\in L^2(\Omega)\). If \(n=1\), \(\Omega=(0,1)\) is considered. The functions \(a\) and \(b\) are chosen from certain classes of monotone functions from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) (resp. \(\mathbb{R}^1\) to \(\mathbb{R}^1\)). An optimization theoretic approach and an algorithm for the estimation of state-dependent coefficients \(a\) and \(b\) are presented. The identification problem is formulated as a nonlinear least-squares problem, which is approximated and analyzed in the framework of convex analysis. The algorithm is based on a nonlinear control formulation of the modified least-squares approach.
Reviewer: G.Dimitriu (Iaşi)

MSC:

35R30 Inverse problems for PDEs
49K20 Optimality conditions for problems involving partial differential equations
49M05 Numerical methods based on necessary conditions
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[1] H. T. Banks and K. Kunisch: Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, 1989. · Zbl 0695.93020
[2] V. Barbu and Th. Precupanu: Convexity and Optimization in Banach Spaces, Reidel, Dordrecht, 1986. · Zbl 0594.49001
[3] H. Brezis: Problémes unilateraux, J. Math. Pures Appl., 51 (1972), 1-64.
[4] G. Chavent and J. Jaffrè: Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. · Zbl 0603.76101
[5] G. Chavent and K. Kunisch: A geometric theory forL 2-stability of the inverse problem in a one-dimensional elliptic equation from an H1 observation, Appl. Math. Optim., 27 (1993), 231-260. · Zbl 0776.35077
[6] R. Ewing, ed.: The Mathematics of Reservoir Simulation, SIAM, Philadelphia, PA, 1983. · Zbl 0533.00031
[7] P. Grisvard: Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. · Zbl 0695.35060
[8] K. Ito and K. Kunisch: The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control Optim., 28 (1990), 113-136. · Zbl 0709.93021
[9] O. A. Ladyzhenskaya and N. N. Ural’tseva: Equations à Dérivés Partielles de Type Elliptique, Dunod, Paris, 1968.
[10] W. W. Yeh: Review of parameter identification procedures in groundwater hydrology: the inverse problem, Water Resources Res., 22 (1986), 95-108.
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