Random Walks in Frequency and the Reconstruction of Obstacles with Cavities from Multi-frequency Data

Abstract

Inverse obstacle scattering is the recovery of an obstacle boundary from the scattering data produced by incident waves. This shape recovery can be done by iteratively solving a PDE-constrained optimization problem for the obstacle boundary. While it is well known that this problem is typically non-convex and ill-posed, previous investigations have shown that in many settings these issues can be alleviated by using a continuation-in-frequency method and introducing a regularization that limits the frequency content of the obstacle boundary. It has been recently observed that these techniques can fail for obstacles with pronounced cavities, even in the case of penetrable obstacles where similar optimization and regularization methods work for the equivalent problem of recovering a piecewise constant wave speed. The present work investigates the recovery of obstacle boundaries for impenetrable, sound-soft media with pronounced cavities, given multi-frequency scattering data. Numerical examples demonstrate that the problem is sensitive to the choice of iterative solver used at each frequency and the initial guess at the lowest frequency. We propose a modified continuation-in-frequency method which follows a random walk in frequency, as opposed to the standard monotonically increasing path. This method shows some increased robustness in recovering cavities, but can also fail for more extreme examples. An interesting phenomenon is observed that while the obstacle reconstructions obtained over several random trials can vary significantly near the cavity, the results are consistent for non-cavity parts of the boundary.

Appendices

A Forward Scattering Problem and Fréchet Derivatives

The solution of the forward scattering problem (1), relies on the use of the Helmholtz single and double layer potentials given by

$$\begin{aligned} \mathcal {S}[\mu ](\varvec{x}) = \int _{\Gamma } G(k |{\varvec{x}}-{\varvec{y}}|) \mu ({\varvec{y}})ds(\varvec{y}) \,, \quad \mathcal {D}[\sigma ](\varvec{x}) = \int _{\Gamma } \frac{\partial G(k |{\varvec{x}}-{\varvec{y}}|)}{\partial \nu (\varvec{y})} \sigma ({\varvec{y}})ds(\varvec{y}) \,. \end{aligned}$$

Here $G(r)=i H^{(1)}_{0}(kr)/4$ is the Green’s function for the Helmholtz equation with wave number k, with $H_{0}^{(1)}(z)$ being the Hankel function of the first kind of order zero. The scattered field $u^\text {scat}$ is then represented as $u^\text {scat}= (\mathcal {D}+ ik \mathcal {S})[\sigma ](\varvec{x})$ for some unknown density $\sigma$. Since the layer potentials satisfy the Helmholtz equation in $\mathbb {R}^{2} {\setminus } \Gamma$, along with the Sommerfeld radiation condition at $\infty$, the density $\sigma$ is determined by enforcing the boundary conditions. Using the jump relations satisfied by the layer potentials [22], we obtain that $\sigma$ must satisfy the following integral equation,

$$\begin{aligned} \frac{\sigma (\varvec{x})}{2} + D^{PV}[\sigma ](\varvec{x}) + ik S[\sigma ](\varvec{x}) = -u^\text {inc}(\varvec{x}) \,, \quad \varvec{x}\in \Gamma \,, \end{aligned}$$

(26)

where S, and $D^{\text {PV}}$ are restrictions of the single layer potential and the principal value of the double layer potential to the boundary $\Gamma$.

Equation (26) is discretized using the Nyström method. The boundary is discretized with equispaced nodes in the arclength parametrization, and the weakly singular layer potentials are evaluated using a $16^{th}$ order Alpert correction [1]. For the range of problems considered in this work, the number of points on the boundary, N, is typically less than $10^{4}$, and hence the discretized system of equations are solved directly using Gaussian elimination. For higher frequencies, and more complicated geometries which lead to significantly larger values of N, it is desirable to use a fast direct solver for obtaining an approximate inverse, which can be computed in $O(N \log ^{p} N)$ time, see for example [2, 3, 15,16,17, 24, 25, 28, 33, 34]. After computing the density $\sigma$, the scattered field at the receptors can be evaluated using the trapezoidal rule which is spectrally accurate since the receptors are typically far-away from the obstacle boundary.

Remark 4

The preference for a fast direct solver over the use of iterative solvers accelerated with FMMs is two fold: first, N is typically large at large frequencies for which the linear system tends to be illconditioned and require O(k) iterations where k is the wavenumber, and secondly, solution to the same linear system is required for $N_{d}$ right hand sides for evaluating the loss function, and $N_{d} N_{f}$ different boundary data for evaluating the Fréchet derivative along $N_{f}$ directions.

We now turn our attention to the evaluation of the Fréchet derivative required for computing the Gauss–Newton/Steepest descent update when solving the single frequency inverse problem (5). In a slight abuse of notation, suppose now that the measurements are made for a single incident direction, $u^\text {inc}= \exp {ik \varvec{x}\cdot d}$. The Fréchet derivative for the case of multiple incident directions can be computed by appropriately stacking the Fréchet derivatives for single incident directions similar to (2). Let $\gamma$ as before denote the parametrization of $\Gamma$. Then the Fréchet derivative in the direction $\gamma _{u}$ denoted by $\mathcal {J}_{\gamma } \cdot \gamma _{u}$ is the potential v evaluated at the receptor locations, where v is the solution to

$$\begin{aligned} \begin{aligned}&\Delta v+k^2 v = 0, \quad \text {in} ~\mathbb {R}^{2} {\setminus } \Omega , \\&v = - (\nu \cdot \gamma _{u}) \frac{\partial u^{\text {tot}}}{\partial \nu } \, \quad \text {on} ~\Gamma , \\&{{\displaystyle \lim _{|{\varvec{x}}|\rightarrow \infty }\;}} |{\varvec{x}}|^{1/2}\left( \frac{\partial v}{\partial r} - ik v \right) = 0, \end{aligned} \end{aligned}$$

(27)

where $u^{\text {tot}} = u^\text {inc}+ u^\text {scat}$, and $\nu$ as before is the outward normal to $\Gamma$. Note that v satisfies exactly the same PDE as $u^\text {scat}$ and hence the same integral formulation, discretization, and solution operator can be applied to evaluate the solution v, reiterating the advantage of using a direct solver or fast direct solver for approximating the solution operator.

Remark 5

The evaluation of the normal derivative of $u^\text {scat}$ on $\Gamma$ required for the evaluation of the boundary data for v can pose a challenge, owing to the need for evaluating the normal derivative of the double layer potential on the boundary which has a hypersingular kernel. While there are a vast variety of quadrature methods that one could use to evaluate the data on the boundary such as [26, 29, 31, 32, 37, 38], the normal derivative can also be evaluated as the solution of the following integral equation

$$\begin{aligned} (I/2+S^\prime -ikS)\frac{\partial u^{\text {tot}}}{\partial \nu }(x)=\frac{\partial u^\text {inc}}{\partial \nu }(x)-ik u^\text {inc}(x) \,, \end{aligned}$$

(28)

see [22], for example. Here $S^\prime$ is the normal derivative of the operator S which is weakly singular like D, and S on the boundary, thereby avoiding the necessity of evaluating hypersingular integral operators on the boundary.

B Linear Sampling Method (LSM)

The LSM was first introduced in [21], wherein the level-set of an appropriate indicator function evaluated on a region containing the support of the domain is used for the solution of the inverse obstacle scattering problem. While, the LSM was originally developed when measurements of the far-field pattern are made, it’s extension to the case of distant scattered field measurements is straight-forward.

Let $u^\text {scat}(\varvec{x},\theta )$ denote the scattered field at x generated by the scattering of the incident plane wave $u^\text {inc}(\varvec{x}) = \exp {(i k \varvec{x}\cdot (\cos {(\theta ), \sin {(\theta )}}))}$, and let $\mathcal {L}$ denote the operator given by

$$\begin{aligned} \mathcal {L}[g](\varvec{x}) = \int _{0}^{2\pi } u^\text {scat}(\varvec{x},\theta ) g(\theta ) d\theta \,, \end{aligned}$$

(29)

where g is known as the Herglotz wave function. In particular, $\mathcal {L}[g](\varvec{x})$ is the solution to the Helmholtz equation with Dirichlet boundary conditions, and an incident field given by

$$\begin{aligned} u^\text {inc}(\varvec{x}) = \int _{0}^{2\pi } \exp {(i k \varvec{x}\cdot (\cos {(\theta ), \sin {(\theta )}}))} g(\theta ) d\theta \,. \end{aligned}$$

(30)

Let $G(k(|\varvec{x}-\varvec{y}|)$ as before denote the Green’s function for the Helmholtz equation with wave number k, and for each $\varvec{x}\in \mathbb {R}^{2}$, let $g(\varvec{x},\theta )$ denote the Herglotz wave function satisfying

$$\begin{aligned} A \cdot \begin{bmatrix} g(\varvec{x},\theta _{1}) \\ g(\varvec{x},\theta _{2}) \\ \vdots \\ g(\varvec{x},\theta _{N_{d}}) \\ \end{bmatrix} = \begin{bmatrix} G(k|\varvec{x}-\varvec{x}_{1}|) \\ G(k|\varvec{x}-\varvec{x}_{2}|) \\ \vdots \\ G(|\varvec{x}-\varvec{x}_{N_{t}}|) \end{bmatrix} \,, \end{aligned}$$

(31)

where

$$\begin{aligned} A =\frac{\sqrt{8\pi } \exp {(-i\pi /4)}}{\sqrt{k} N_{d}} \begin{bmatrix} u^\text {scat}(\varvec{x}_{1}, \theta _{1}) &{}\quad u^\text {scat}(\varvec{x}_{1},\theta _{2}) &{}\quad \ldots &{}\quad u^\text {scat}(\varvec{x}_{1},\theta _{N_{d}}) \\ u^\text {scat}(\varvec{x}_{2}, \theta _{1}) &{}\quad u^\text {scat}(\varvec{x}_{2},\theta _{2}) &{}\quad \ldots &{}\quad u^\text {scat}(\varvec{x}_{2},\theta _{N_{d}}) \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ u^\text {scat}(\varvec{x}_{N_{t}}, \theta _{1}) &{}\quad u^\text {scat}(\varvec{x}_{N_{t}},\theta _{2}) &{}\quad \ldots &{}\quad u^\text {scat}(\varvec{x}_{N_{t}},\theta _{N_{d}}) \\ \end{bmatrix} \,. \end{aligned}$$

(32)

Here $g(\varvec{x},\cdot )$ is the Herglotz wave function which reproduces the field due to a point source located at $\varvec{x}\in \mathbb {R}^{2}$ at the receptor locations $\varvec{x}_{1}, \varvec{x}_{2} \ldots \varvec{x}_{N_{t}}$. In the limit of number of receptors, and incident directions going to $\infty$, the norm of the Herglotz wave function for $\varvec{x}$ inside the obstacle tends to $\infty$, while it remains finite for $\varvec{x}$ outside the domain. Thus, any function of the norm of the Herglotz wave function can be used as a test function for estimating the boundary of the obstacle. In practice, typically the following function of the Herglotz wave function is used as an indicator function,

$$\begin{aligned} h(\varvec{x}) = \log { \left( \sqrt{\sum _{\ell =1}^{N_{d}} |g(\varvec{x},\theta _{\ell })|^2} \right) } \,, \end{aligned}$$

(33)

where the boundary of the obstacle is defined as a level set of h, i.e. $\tilde{\Gamma }_{0} = \{\varvec{x}: \, h(\varvec{x}) = C\}$.

The computation of $h(\varvec{x})$ requires the solution to the linear system in (31), for every $\varvec{x}$ in the domain of interest. However, the system of equations tends to be extremely ill-conditioned and typically the Herglotz wave functions are computed via the solution of the following Tikhonov-regularized problem:

$$\begin{aligned} \varvec{g}(\varvec{x},\cdot ) = \min _{\varvec{g}} \Vert A\cdot \varvec{g}(\varvec{x}, \cdot ) - \varvec{G}_{\varvec{x}} \Vert ^2 + \alpha ^2 \Vert \varvec{g}(\varvec{x},\cdot ) \Vert ^2 \,, \end{aligned}$$

(34)

where $\alpha$ is the Tikhonov-regularization parameter, $\varvec{G}_{\varvec{x}} = [G(k|\varvec{x}-\varvec{x}_{1}|)\, ; G(k|\varvec{x}-\varvec{x}_{2}|)\,; \ldots G(k|\varvec{x}-\varvec{x}_{N_{t}}|)]$, and $\varvec{g}(\varvec{x},\cdot ) = [g(\varvec{x},\theta _{1})\,; g(\varvec{x},\theta _{2})\,; \ldots g(\varvec{x},\theta _{N_{d}})]$. Finally, owing to the equispaced tabulation of the function $h(\varvec{x})$, the level set $h(\varvec{x}) = C$, computed numerically through standard contour extractors tends to be non-smooth. We smoothen the initial guess by approximating it with a star-shaped obstacle of the form $r(t)(\cos {(t)}, \sin {(t)})$, with

$$\begin{aligned} r(t)=c_0+\sum _{n=1}^N\left( c_n\cos (nt)+c_{n+N}\sin (nt)\right) \,, \quad t\in [0,2\pi ) \,. \end{aligned}$$

(35)

Let $\varvec{t}_{\ell }$, $\ell =1,2,\ldots M$ denote each point in the level set, and let $\phi _{\ell }.= \text {Arg}(\varvec{t}_{\ell }) \in [0,2\pi )$. Then the coefficients $[c_{0}; \ldots c_{2N}]$ are obtained via the least-square solution of the following system of equations

$$\begin{aligned} c_{0} + \sum _{n=1}^{N} \left( c_n\cos (n \phi _{\ell })+c_{n+N}\sin (n \phi _{\ell })\right) = |\varvec{t}_{\ell }| \,. \end{aligned}$$

(36)

For the examples used in this paper, we set $\alpha = 10^{-3}$, and $N=10$, unless stated otherwise.