Statistical Inverse Estimation Of Source Terms In The Helmholtz Equation

Helmholtz equation is a time-harmonic form of the wave equation,which is widely used in electromagnetic radiation,seismology,and acoustics in physics.For example,the propagation and scattering of acoustic and electromagnetic waves,noise control,structural vibration,and cavity scattering can all be described by the Helmholtz equation.The study of the Helmholtz equation is the basis of scattering theory,and the identification of its source term is an important part of the inverse scattering problem.Therefore,the research on how to invert the source term of the Helmholtz equation according to the boundary measuring points has a profound practical application background.In this thesis,the identification of the source term of the Helmholtz equation in two different regions is mainly discussed in the homogeneous medium.Since the source term discontinuously depends on the measured data,that is,the small error perturbation of the measured data will lead to the solution blowup.This thesis mainly discusses the problem by using a statistical inverse estimation method.When the measurement data is random and contains a random error,the estimation formula of the source term is obtained,and the error estimation between the approximate solution and its exact solution is proved.Numerical experiments show that this statistical inverse estimation method is effective and can give stable numerical inversion results.The first chapter briefly introduces the physical background of the Helmholtz equation and the development of the source term identification problem and expounds the statistical inverse estimation method used in this thesis and the main work.In chapter 2 and chapter 3,the unknown source identification problem of the Helmholtz equation in a two-dimensional semi-zonal region and three-dimensional bounded cylindrical region is considered respectively.Then,through the observation data samples on the boundary,the estimation formula of the source term on the two different regions is obtained using the statistical inverse estimation method,and the error estimation of the exact solution and its approximate solution is proved.In chapter 4,numerical examples are given to verify the effectiveness of the proposed method in two different regions.