Regularization Method For Solving Acoustic Scattering Problems And Abel Integral Equation

Since the 1960's, the inverse problems of "finding the cause or inverse-image(input) from known effect or behavior(output)" , which named as "the inverse problems of the mathematics and physics" , have been proposed in many research fields of science and technology. The inverse acoustic scattering problem is the typical one of those inverse problems, and has many important uses in all kinds of fields, such as undamaged inspection, medicine imaging, remote sensing, tracking radar and resources exploration of petroleum, mineral and ocean. It has been developed to be a cross-subject of numerical and applied mathematics and system science. In this dissertation, the author discusses the numerical solution for the direct acoustic scattering problem with the Neumann boundary condition, and pays more attentions to discuss numerical recovering the shape of the obstacle region, which is the inverse problem of the direct acoustic scattering problem with the Dirichlet or impedance boundary condition. Besides the above discussions, based on the Tikhonov regularized method, the author puts forward another numerical inversion algorithm for numerical solution of a class of Abel integral equations. The main works of this dissertation are as follows:1. The numerical algorithms of the direct acoustic scattering problem with the Neumann boundary condition is studied. This method transform the original problem to the second kind Fredholm boundary integral equation which is solved easily by Nystr(o|¨)m, or Collocation, or Galerkin methods, and to the first kind Fredholm boundary integral equation which is solved by the Tikhonov regularized method. The numerical experiments prove those numerical methods be effective, and the results can be used as valid data in the study of the inverse acoustic scattering problem.2. Tikhonov regularization numerical method for recovering the shape of an obstacle from the information of time harmonic incident acoustic wave and far field pattern of the scattered wave with frequency in the resonance region is investigated. Firstly, the indicator function which representing the scattering characteristics is constructed. Secondly, based on the property of the function, the fundamental equation for solving the inverse problem is established. Then, the shape of the obstacle is determined by numerical solution. This method requires no a prior information of the type of boundary condition imposed on the objects and its shape. Numerical experiments show the validity and practicality of this method.3. A scheme of linear sampling method is developed for the inverse problem which recovering the shape of an obstacle from the information of time harmonic incident acoustic wave and the far field pattern of scattered wave with frequencyin the resonance region. This algorithm is based on solving a linear integral equation of the first kind, which avoids the use of nonlinear optimization methods and requires no a priori information at all about the geometry of the scattering obstacle and the type of boundary conditions imposed on the objects. Numerical experiments indicate this algorithm has more efficiency and numerical stability.4. Based on Tikhonov regularization theory and the theoretical inversion formula of Abel integral equations, a numerical inversion algorithm is proposed by the regularized treatment on numerical differentiation and the weighted Gauss-type integral. Theoretical analysis and numerical experiments prove this algorithm possess high accuracy and better numerical stability.