Numerical Estimation of the Inverse Eigenvalue Problem for a Weighted Helmholtz Equation

The inverse eigenvalue problem for a weighted Helmholtz equation is investigated. Based on the finite spectral data, the density function is estimated. The inverse problem is formulated as a least squared functional with respect to the density function, with a L^2 regularity term. The continuity of the eigenpairs with respect to the density is proved. Mathematical properties of the continuous and the discrete optimization problems are established. A conjugate gradient algorithm is proposed. Numerical results for 1 D and 2 D inverse eigenvalue problem of the weighted Helmholtz equation are presented to illustrate the effectiveness and efficiency of the proposed algorithm.