## 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
Full Text:

### References:

 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.