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Propagation of singularities for linearised hybrid data impedance tomography. (English) Zbl 1433.65262

Summary: For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure based of the least squares finite element method is derived, and a series of numerical experiments visualise exactly how the loss of ellipticity manifests itself as propagating singularities.

MSC:

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C55 Biomedical imaging and signal processing
35S15 Boundary value problems for PDEs with pseudodifferential operators
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

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References:

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