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| On a Hybrid Method for InverseTransmission Eigenvalue Problems |
| Weishi Yin,Zhaobin Xu,Pinchao Meng,Hongyu Liu |
| (School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun, Jilin 130022, China;Department of Mathematics, City University of Hong Kong, Kowloon,Hong Kong, China) |
| DOI： |
| Abstract: |
| In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of
a penetrable medium scatterer. The linear sampling method is employed to
determine the transmission eigenvalues within a certain wavenumber interval
based on far-field measurements. Based on a prior information given by the
linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals
into several subintervals, ensuring that each transmission eigenvalue is located
in its corresponding subinterval. In each such subinterval, the wavenumber that
yields the maximum value of the indicator functional will be included in the
input set during the generation of the training data. This technique for data
generation effectively ensures the consistent dimensions of model input. The
refractive index and shape are taken as the output of the network. Due to the
fact that transmission eigenvalues considered in our method are relatively small,
certain super-resolution effects can also be generated. Numerical experiments
are presented to verify the effectiveness and promising features of the proposed
method in two and three dimensions. |
| Key words: Inverse transmission eigenvalue problem, linear sampling method, neural
network, super-resolution. |