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Zbl 1195.65159
Orlovsky, D.; Piskarev, S.
On approximation of inverse problems for abstract elliptic problems. (English)
[J] J. Inverse Ill-Posed Probl. 17, No. 8, 765-782 (2009). ISSN 0928-0219; ISSN 1569-3945/e

The authors consider an inverse problem of an abstract elliptic differential equation with respect to a general Banach space. The problem is overdetermined and thus an operator-valued source term has to be identified such that a solution exists. The authors investigate two scenarios. Firstly, a general semidiscretisation yields a sequence of abstract ordinary differential equations (ODEs) of second order. Secondly, the abstract ODEs are discretised by the symmetric difference quotient of second order. The convergence of the approximations is analysed in corresponding Banach spaces, where operator theory as well as the theory of analytic $C_0$-semigroups are applied. Assuming uniformly positive operators in the semidiscretisation, the authors prove the uniform convergence in both scenarios with a constant operator as source term. Furthermore, the authors investigate the case of a class of time-dependent operators as source term. Again the uniform convergence is proved in both scenarios under certain assumptions. The discussion of test examples or numerical simulations are not within the scope of the paper.
[Roland Pulch (Wuppertal)]

MSC 2000:: *65N21 Inverse problems
65J22 Inverse problems
34G10 Linear ODE in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
35R30 Inverse problems for PDE
65N40 Methods of lines (BVP of PDE)

Keywords: abstract differential equations; abstract elliptic problem; analytic $C_0$-semigroups; Banach spaces; semidiscretization; inverse problem; overdetermination; well-posedness; difference schemes; discrete semigroups; general approximation scheme; compact convergence of resolvents

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