New computational methods for inverse wave scattering with a new filtering technique

Abstract

This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a moment problem for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise.

Appendix

To obtain an approximate solution for the Helmholtz equation in Eq. (39) we can rewrite it according to

$$\begin{aligned} \varDelta \xi +k^2\xi + \bar{p}({\mathbf{x}})\xi e=\delta ({\mathbf{x}}-{\mathbf{x}}_0),\xi =0,{\mathbf{x}}\in {\partial }\varOmega , \end{aligned}$$

(57)

where $\bar{p}({\mathbf{x}})={\hat{p}}({\mathbf{x}})-1$. We can then consider a similar perturbation approximation and obtain the solution as a series given by $\xi =\xi ^0+\xi ^1+\xi ^2+\xi ^3+\dots$ where

$$\begin{aligned}&\varDelta \xi ^0+k^2\xi ^0=\delta ({\mathbf{x}}-{\mathbf{x}}_0),\xi ^0=0,{\mathbf{x}}\in {\partial }\varOmega , \end{aligned}$$

(58)

$$\begin{aligned}&\varDelta \xi ^1+k^2\xi ^1=-k^2\bar{p}\xi ^0,\xi ^1=0,{\mathbf{x}}\in {\partial }\varOmega , \end{aligned}$$

(59)

$$\begin{aligned}&\varDelta \xi ^2+k^2\xi ^2=-k^2\bar{p}\xi ^1,\xi ^2=0,{\mathbf{x}}\in {\partial }\varOmega , \end{aligned}$$

(60)

$$\begin{aligned}&\varDelta \xi ^3+k^2\xi ^3=-k^2\bar{p}\xi ^2,\xi ^3=0,{\mathbf{x}}\in {\partial }\varOmega \end{aligned}$$

(61)

Note that $\hat{p}({\mathbf{x}})$ is zero for most of the region, and the above series converges very fast. In all calculations presented here, we are including four terms in the series.