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The inverse eigenvalue problem for a weighted Helmholtz equation. (English) Zbl 1068.65131
Summary: Given the \(m\) lowest eigenvalues, we seek to recover an approximation to the density function \(\rho\) in the weighted Helmholtz equation \(-\Delta u=\lambda\rho u\) on a rectangle with Dirichlet boundary conditions. The density \(\rho\) is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with \(\rho=1\). The matrix inverse problem is solved by a fixed-point iterative method and a density function \(\rho^*\) is constructed which has the same \(m\) lowest eigenvalues as the unknown \(\rho\). The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.

MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R30 Inverse problems for PDEs
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