Kuo Ming Lee

Integral equation methods for scattering from an impedance crack

For the scattering problem for time-harmonic waves from an impedance crack in two dimensions, we give a uniqueness and existence analysis via a combined single- and double-layer potential approach in a Holder space setting leading to a system of integral equations that contains a hypersingular operator. For its numerical solution we describe a fully discrete collocation method based on trigonometric interpolation and interpolatory quadrature rules including a convergence analysis and numerical examples.

Inverse scattering via nonlinear integral equations for a Neumann crack

In this paper we present a new method for solving the time-harmonic acoustic inverse scattering problem for a sound-hard crack in . Using the integral equation method to solve the inverse scattering problem, one obtains a Fredholm integral equation of the first kind. Instead of applying regularized Newtons method directly to this integral equation, we derive an equivalent system of two nonlinear integral equations for the inverse problem. In this setting, not only can the regularized Newtons method still be used to solve the inverse problem numerically, but also has the advantage of removing the need to solve a related direct problem at every iteration.

A second degree Newton method for an inverse obstacle scattering problem

Abstract A regularized second degree Newton method is proposed and implemented for the inverse problem for scattering of time-harmonic acoustic waves from a sound-soft obstacle. It combines ideas due to Johansson and Sleeman [18] and Hettlich and Rundell [8] and reconstructs the obstacle from the far field pattern for scattering of one incident plane wave.

A two step method in inverse scattering problem for a crack

In this paper, we present a method for solving a time-harmonic acoustic inverse scattering problem for a sound-soft crack in R2. Based on the integral equation method, our method splits the nonlinear, severely ill-posed inverse problem into a linear well-posed direct problem and a nonlinear ill-posed problem. At this setting, not only the regularized Newton’s method can still be used to solve the inverse problem numerically but also with the advantage of keeping the structure of the algorithm simple and efficient.

Inverse Scattering from a Sound-Hard Crack via Two-Step Method

We present a two-step method for recovering an unknown sound-hard crack in ℝ 2 from the measured far-field pattern. This method, based on a two-by-two system of nonlinear integral equations, splits the reconstruction into two consecutive steps which consists of a forward and an inverse problems. In this spirit, only the latter needs to be regularized.

An inverse scattering problem from an impedance obstacle

In this paper we consider an inverse scattering problem from an obstacle with impedance boundary condition. Our aim is to recover the unknown scatterer from the far field pattern iteratively assuming the impedance function. Our method, while remaining in the framework of Newtons method, based on a system of two nonlinear integral equations which is equivalent to the original inverse problem, avoids the need of calculating a direct problem at each iteration. Because of the ill-posedness of this problem, regularization method for example, Tikhonov regularization, is incorporated in our solution scheme. Several numerical examples with only one incident wave are given at the end of the paper to show the feasibility of our method.

A second degree Newton method in inverse scattering problem for a crack

For the inverse scattering problem from a sound-soft crack a second degree Newton method is proposed in this paper. Based on integral equations method, our scheme splits the inverse problem into a well-posed problem and a nonlinear ill-posed problem which will be linearized using a second degree method. Some reconstructions will be given to demonstrate the feasibility of the proposed method.

Inverse scattering problem from an impedance obstacle via two-steps method

In this paper we deal with the inverse scattering problem for an impedance obstacle. The aim is to recover both the impedance function and the scatterer simultaneously. Based on boundary integral equations, our method splits the inverse problem into a well-posed direct problem followed by a smaller ill-posed problem which has advantages both in understanding the inverse problem and in the numerical reconstructions.