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Institute for Applied and Numerical Mathematics

Research Group 4: Inverse Problems

Research

Research Areas

Direct and Inverse Scattering Problems

Work Group Inverse Problems
Prof. Dr. A. Kirsch, Dr. T. Arens, PD. Dr. F. Hettlich

Scattering problems

Wave phenomena occur in various situations for instance as acustic waves at sound sources, as electromagnetic waves in tele communications or as elastic waves propagating from earthquakes. A common feature of all of these waves is, that they will be scattered, absorbed and/or transmitted by different media, by scattering objects'.

The mathematical modelling is done by partial differential equations like the wave equation
$U_{tt}(x,t)= c\,\Delta U(x,t),$
with spatial variable $x$ and time variable $t$ . A plane monochromatic wave with frequency $\omega>0$ and incident direction $\hat{\theta}$ has the form
$U^{ein}(x,t;\hat{\theta})\ =\cos(kx\cdot\hat{\theta}-\omega t)\ =\ \Re e^{i(kx\cdot\hat{\theta}-\omega t)}$
where $k=\omega/c$ denotes the wave number and $c>0$ the velocity in the medium.

The scattering of $U^{ein}$ leads to a scattered field $U^s=U^s(x,t;\hat{\theta})$ and the total field $U=U^{ein}+U^s$ as a superposition of the incident and the scattered wave. If the scattering obstacle is constant in time, it holds
$U^s(x,t;\hat{\theta})\ =\ \mathrm{Re}\left[u^s(x;\hat{\theta})\,e^{-i\omega t}\right],\quad U(x,t;\hat{\theta})\ =\ \mathrm{Re}\left[u(x;\hat{\theta})\,e^{-i\omega t}\right],$
where $u^s$ and $u$ satisfy the reduced wave equation, or Helmholtz equation $\Delta u\ +\ k^2u\ =\ 0$ , in the exterior of the scattering obstacle. Assuming a bounded scattering object the scattered field will be of the asymptotic form
$\displaystyle u^s(x;\hat{\theta})\ \approx\ \frac{\exp(ikr)}{4\pi\,r}\,u^\infty(\hat{x};\hat{\theta})$
in large distance to the obstacle. Here $r=|x|$ and $\hat{\theta}=x/|x|$ denote the polar coordinates of $x$ .

$Two far field patterns$ The picture shows level lines of far field pattern $u^\infty_1$ and $u^\infty_2$ with respect to the angles given by $\hat{\theta}$ and $\hat{x}$ in $0^o$ to $360^o$ (for two dimensional examples).

Which object belongs to which far field pattern?

The direct scattering problem consists in showing existence, uniqueness, stability, and the numerical computation (and its graphical illustration) of the fields, if the scattering object is known.
The invers scattering problem consists in the identification of the scattering object, if far field patterns $u^\infty(\hx,\hat{\theta})$ for directions $\hat{x},\hat{\theta}$ are known by measurements (i.e. if such plots are given).

Research of the work group

The direct problems can be written as integral equations using fundamental solutions, which can be helpful for theoretical questions on existence and stability as well as for numerical approximations. Currently in the working group algorithms for numerical solution of three dimensional scattering problems in case of biperiodic gratings are developed and investigated.

The fast and efficient evaluation of solutions of the direct problems is essential for using iterative methods (see below) in solving the corresponding inverse problems. Additionally, by the direct solver we obtain simulated data, which can be used to verify and compare various inversion schemes. The influence of parameter, which describe the shape or the material, can be analysed systematically by such data.

Iterative Methods: Already in case of the scattering of only one incident field at an obstacle $D$ the inverse problem can be considered. It consists in solving a non-linear and ill posed equation of the form
${\cal F}(\partial D)=u_\infty(.,\hat{\theta}).$
Iterative Newton type methods like
$\big({({\cal F}'[D_j])^*}}\,{\cal F}'[D_j] +\alpha I\big)}h\ =\ ({\cal F}'[D_j])^*} \big(u_\infty(.,\hat\theta}) - {\cal F}(D_j)\big)$
will require the domain derivative ${\cal F}'$ of the operator. Additionally a regularization is necessary, since otherwise the linearised equation has no stable inversion. By stopping the iteration early enough a good reconstruction from a far field pattern is possible. The picture shows the best, a mean, and the worst reconstruction from $100$ experiments with $10\%,$ random noise in the data and with $k=1$ computed by a modified regularization scheme, which uses also the second domain derivative.

$center$

The factorization method: Iterative algorithm are all-purpose methods and lead still to the best reconstruction results. But they cost a high computational effort (in any step a direct boundary value problem has to be solved) and admit at most local convergence. Therefore, recently different approaches were established, which avoid these drawbacks. Various reconstruction algorithms based on the factorization method have been developed in the work group.

The far field pattern $u^\infty=u^\infty(\hat{x};\hat{\theta})$ gives the kernel of the far field operator $F:L^2(S^2)\to L^2(S^2)$ , defined by
$(Fg)(\hat{x})\ =\ \int_{S^2}u^\infty(\hat{x};\hat{\theta})\,g(\hat{\theta})\,do(\hat{\theta})\,,\quad \hat{x}\in S^2\,,$
with the unit sphere $S^2$ im $\mathbb{R}^3$ . The far field operator $F$ is compact and normal in case of non-absorbing scattering objects (i.e. $F^\ast F=F\,F^\ast$ ) and can be factorised in the form
$F\ =\ A^\ast T\,A$
with certain operators $A$ and $T$ , where $A$ is compact and $T$ satisfies a Garding inequality. This is considerably used in representing the characteristic function $\chi_D$ of the set $D$ in the form
$\chi_D(z)\ =\ \operatorname{sign}\left[\sum_{j=1}^\infty \frac{|(\psi_j,\phi_z)_{L^2(S^2)}|^2}{|\lambda_j|}\right]^{-1},$
where $\phi_z(\hat{x})=\exp(-ik\,\hat{x}\cdot z)$ , $\hat{x}\in S^2$ , and a spectral system $\{\lambda_j,\psi_j:j\in\mathbb{N}\}$ of $F$ .

The following reconstructions are comupted by the factorization method from $u^\infty-$ data: (The picture with two obstacles corresponds to the lower one of the two presented far field pattern above.)

$Some reconstructions from inverse scattering problems$

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Department of Mathematics,
Institut for Algebra and Geometry,
Work Group Inverse Problems