Reconstruction Of The Density Function Of The Two-dimensional Helmholtz Equation

Recently, inverse spectral problem has become a very active research direc-tion in the study of inverse problems. There have been a lot of research results about one-dimensional inverse spectral problem. This dissertation mainly focus on the two-dimensional inverse spectral problem for Helmholtz equation. For the reconstruction of unknown symmetric density function p, we firstly intro-duced the Rayleigh-Ritz method which proposed in [(?)]. Then inspired by this approach, we propose a new solving method. Based on the symmetry property of p, we choose n Fourier-type functions as basic functions for the expansion of p. By substituting the approximate expression into the equation, the continuous in-verse eigenvalue problem is converted to an unknown coefficient solving problem of the above mentioned basic functions. Then a least squares problem is formu-lated for solving the above problem. Taking advantage of the sensitivity analysis of the eigenvalue which discretized by rectangle finite element, the solution of the least squares problems via the steepest descent method is discussed and then an approximation to the unknown density is recovered. Numerical results show our method is feasible.