Abstract
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 1D and 2D inverse eigenvalue problem of the weighted Helmholtz equation are presented to illustrate the effectiveness and efficiency of the proposed algorithm.
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Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Funding
This research is supported by Natural Science Foundation of Zhejiang Province, China (No. LY21A010011) and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Cutie Grand Agreement (No. 823731 CONMECH).
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All authors contributed to the study conception and algorithm design. The theoretical part is derived mainly by ZZ and XC. The numerical part is mainly performed by XG. The first draft of the manuscript was written by XG and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Zhang, Z., Gao, X. & Cheng, X. Numerical Estimation of the Inverse Eigenvalue Problem for a Weighted Helmholtz Equation. J Sci Comput 96, 16 (2023). https://doi.org/10.1007/s10915-023-02242-1
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DOI: https://doi.org/10.1007/s10915-023-02242-1