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Reconstruction of an impedance in two-dimensions from spectral data. (English) Zbl 1081.35145
Summary: The two-dimensional spectral inverse problem involves the reconstruction of an unknown coefficient in an elliptic partial differential equation from spectral data, such as eigenvalues. Projection of the boundary value problem and the unknown coefficient onto appropriate vector spaces leads to a matrix inverse problem. Unique solutions of this matrix inverse problem exist provided that the eigenvalue data is close to the eigenvalues associated with the analogous constant coefficient boundary value problem. We discuss here the application of such a technique to the reconstruction of an impedance \(p\) in the boundary value problem \[ -\nabla(p \nabla u)=\lambda pu\text{ in }R,\quad u=0\text{ on }\delta R, \] where \(R\) is a rectangular domain. The matrix inverse problem, although nonstandard, is solved by a fixed-point iterative method and an impedance function \(p^*\) is constructed which has the same \(m\) lowest eigenvalues as the unknown \(p\). Numerical evidence of the success of the method is presented.

MSC:
35R30 Inverse problems for PDEs
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
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