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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 \(L^ 2\).

Reviewer: B.Hofmann (Chemnitz)

### MSC:

35R30 | Inverse problems for PDEs |

34A55 | Inverse problems involving ordinary differential equations |

### Keywords:

size \(x\) curvature; diffusion coefficient in two-point boundary value problems; stability theory of Guy Chavent; output least-squares for nonlinear problems; nonconvex attainable set
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\textit{G. Chavent} and \textit{K. Kunisch}, Appl. Math. Optim. 27, No. 3, 231--260 (1993; Zbl 0776.35077)

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### References:

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