Numerical Method For Inverse Sound-hard Obstacle Scattering Problem With Phaseless Far-field Data

In this paper,we concider the inverse scattering problem of determining the shape of a two-dimensional sound-hard obstacle from phaseless far-field data,that is,determining the shape of a sound-hard obstacle by using the modulus of the far-field data for a single incident plane wave.We firstly consider solving the forward scattering problem to obtain the data which is needed for the inverse problem: 1.By using the layer approach,boundary condition and jump relation,the original problem is transformed into the problem of solving boundary integral equation.2.We describe a fully discrete method based on the trigonometric differential for solving the density function from the hypersingular integral equation which is arised from the combined double-and single-layer approach for the solution of two-dimensional sound-hard scattering problem(i.e.the exterior Neumann problem for the two-dimensional Helmholtz equation).3.The density function is introduced into the scattering field or far-field expression of single-and double-layer potential combination form to obtain the direct problem data.Then we give the numerical experiment results to verify the rationality of the discrete method.Finally,we give the corresponding numerical method of the inverse scattering problem with phaseless data: 1.A single-layer potential function is constructed to represent the scattering solution of the inverse problem.By using the potential theory,boundary condition and jump relation,the problem is transformed into a set of boundary integral equations composed of fieldequation and phaseless far-field data equation.The iterative scheme and numerical discretization of the inverse problem are given and the ill posedness of the inverse problem is solved by Tikhonov regularization method.The feasibility of the method is verified by numerical examples.2.The scattering solution of the inverse problem is represented by a function in the form of double-layer potential.The iterative and numerical discretization schemes of the inverse problem are given by using the discretization method for the hypersingular operator given in the direct problem,and the ill posedness of the inverse problem is solved by Tikhonov regularization method.The feasibility of the method is verified by numerical examples.3.The scattering solution of the inverse problem is represented by a function in the form of double and single-layer potential.The iterative and numerical discretization schemes of the inverse problem are given by using the discretization method given in the direct problem,and the ill posedness of the inverse problem is solved by Tikhonov regularization method.The feasibility of the method is verified by numerical examples.