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A geometric theory for \(L^ 2\)-stability of the inverse problem in a one-dimensional elliptic equation from an \(H^ 1\)-observation. (English) Zbl 0776.35077

The stability of estimating the diffusion coefficient in two-point boundary value problems from noisy data of the state observation is considered. This is a new application of the well-known geometrically motivated abstract stability theory of Guy Chavent concerning output least-squares for nonlinear problems. Specifically, it is based on an analysis of the projection of observations on the nonconvex attainable set. The authors define a nonlinear least-squares problem \(Q\)-well-posed in a local sense if the problem has a unique solution, the functional to be minimized has no local minima, any minimizing sequence converges to the solution and the data-to-solution mapping is locally Lipschitz continuous. Under so-called size \(x\) curvature conditions there are formulated theorems that show the \(Q\)-well-posedness in the case of elliptic parameter estimation problems between the spaces \(H^ 1\) and \(L^ 2\).

MSC:

35R30 Inverse problems for PDEs
34A55 Inverse problems involving ordinary differential equations
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