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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References:

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