The augmented Lagrangian method for parameter estimation in elliptic systems. (English) Zbl 0709.93021

The authors consider the problem of determining the unknown functional coefficient q in the elliptic partial differential equation \[ (1)\quad - div(q \text{grad} u)=f\text{ in } \Omega,\quad u=0\text{ on } \Gamma, \] from an observation z of the solution u, where \(\Omega\) is a bounded domain in \(R^ n\), \(n=1,2,3\), with piecewise smooth boundary \(\Gamma\) and \(f\in H^{-1}\) is given. They propose and analyze a hybrid method which not only combines the output least squares (OLS) and the equation error formulation but also inherits the flexibility of the OLS approach and the quadratic structure of the equation error approach. According to this method the minimizing problem in \(H^ 2:\) \[ \text{ minimize } 2^{-1}| u(q)-z|^ 2_ H+(\beta /2)N(q),\quad over\quad Q_{ad}=\{q\in H^ 2(\Omega);\quad q\geq \alpha \text{ and } | q|_{H^ 2}\leq \gamma \}, \] where u(q) is the solution of (1), is viewed as the following constrained minimization problem: \[ (2)\quad \text{ minimize } F(q,u)=2^{-1}| u-z|^ 2_{H^ 1_ 0}+(\beta /2)N(q), \] subject to \(-\nabla (q\nabla u)=f\) in \(H^{-1}\), \(| q|_{H^ 2}\leq \gamma\), \(\alpha\leq q\) on \(\Omega\), in the two independent variables q and u. This is solved by the following Lagrangian algorithm: minimize a sequence of functionals \[ (3)\quad L_{c_ k}(q,u;\lambda^ k)=F(q,u)+<\lambda^ k,e(q,u)>_{H^ 1_ 0}+(c_ k/2)| e(q,u)|^ 2_{H^ 1_ 0}\quad over\quad q\in Q_{ad}, \] with the multiplier sequence \(\{\lambda^ k\}\) in \(H^ 1_ 0\) given by \(\lambda^{k+1}=\lambda^ k+c_ ke(q_ k,u_ k)\), where the function e: \(H^ 2\times H^ 1_ 0\to H^ 1_ 0\) is given by \(e(q,u)=(-\Delta)^{-1}(\nabla \cdot (q\nabla u)+f)\), and the pair \((q_ k,u_ k)\) minimizes (3). The convergence and the rate of convergence of the pair \((q_ k,u_ k)\) to a solution of (2) as well as that of \(\lambda^ k\) are established.
Reviewer: H.Tanabe


93B30 System identification
35R30 Inverse problems for PDEs
49M29 Numerical methods involving duality
93C20 Control/observation systems governed by partial differential equations
35B37 PDE in connection with control problems (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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