| Electromagnetics is an important research field in mathematical physics.It has been widely applied in various fields of science and engineering,such as biomedicine,target tracking,geophysics,antenna synthesis and optoelectronics.This paper mainly studies the application of the Fourier method in two kinds of electromagnetic inverse problems:the multi-frequency inverse problem of Maxwell’s equations and the Cauchy problem connected with the 2D Helmholtz equation.The inverse source problem is to determine the source function by using the radi-ation field data measured on a surface.At a fixed frequency,the source function cannot be uniquely determined by the measured data on the surface due to the existence of non-radiating sources.In order to obtain a unique solution,it is necessary to impose additional constrains on the source.A commonly used choice of constraints is to pick up the solution with a minimum L2 norm,that is,the minimum energy solution.In order to overcome the difficulties faced by the inverse source problem under sin-gle frequency conditions,many scholars consider the use of multi-frequency measure-ment data to reconstruct the source function.Eller and Valdivia considered the inverse problem of identifying the shape and location of a finitely supported source function from measurements of the acoustic field on a closed surface for an infinite unbounded(chosen)frequencies.This method was extended to the case of the Maxwell’s equation-s by Valdivia.Bao et al.proved that the ill-posedness of the inverse source problems decreases as the frequency increases,and first gave the Lipschitz stability estimate for inverse source problems with multiple frequencies.The iterative method has been proposed to solve the multi-frequency inverse source problem for the Helmholtz equation by Bao et al.Then a numerical method based on the Fourier expansion of the source has been proposed by Zhang et al.This Fourier method has been extended to the far-field cases of acoustic inverse source problem by Wang et al.The Cauchy problem for the Helmholtz equation is ill-posed.The main numerical methods for this problem are as follows:the boundary element method,the moment method,the separated variable method and the meshless method.In this paper,a meshless method is studied.The main idea of the meshless method is as follows:the solution of the Cauchy problem can be approximate by a linear com-bination of the special solutions of the Helmholtz equation,and the coefficients can be calculated by using the known Cauchy data.According to the different selection of the special solutions,there are three kinds of the meshless methods:the fundamental solution method,the boundary knot method and the plane wave method.The numerical methods to solve these two kinds of problems are given,and the error estimate and stability analysis of these algorithms are given.In numerical ex-periments,we give the principles of parameter selection,and verify the feasibility and effectiveness of our methods.The main work of this article is as follows:1.The multi-frequency inverse source problem of the Maxwell’s equations is stud-ied.For the multi-frequency inverse source problem,we consider the source function with the form of J = pf +p×▽g.We first give the decomposition of the polarization vector p,and then use this decomposition to obtain the Fourier expansion of the source function J,so as to get the approximation of J",and obtain the error estimate of this approximation.For the given polarization vector p and measured data,we establish the formulas for calculating the Fourier coefficients and proved the uniqueness of the Fourier coefficients.For the measured data with noise,we calculate the electromagnetic field data on an artificial surface,and get the formulas for calculating the Fourier coefficients.Then we establish the error estimate between the source function and the approximate source function.Finally,through numerical experiments,we verify the effectiveness of our method.2.The Cauchy problem for the 2D Helmholtz equation is studied.We first ap-proximate the solution by using a Herglotz wave function v.The approximation of the density function is obtained by truncating the Fourier series of the density function.As a directly consequence,we can construct vN as the approximation of v.Finally,we prove that the error between v and vN decay exponentially as N tends to infinity.We then obtain the expansion of the plane wave function by using the Taylor series.By truncating this expansion,we can define vN,M to approximate vN.Finally,we prove that the error between vN,m and vN is factorial attenuation as M tends to infinity.We also presented the calculation formulas of vN,M.We emphasize that all integrals in the formulas can be calculated in advance.Due to the Cauchy problem is ill-posed,we use the Tikhonov regularization method with Morozov’s discrepancy principle to solve the corresponding operator equation.We get the error estimate and the result of stability.Finally,in the numerical experiment,we give the principle to choose N and M.The numerical experiments show that the algorithm is fast and effective. |
