Sound-soft Obstacle Research Articles

Accurate computation of scattering poles of acoustic obstacles with impedance boundary conditions

We propose a computation method for scattering poles of impedance obstacles. Boundary integral equations are used to formulate the problem. It is shown that the scattering poles are the eigenvalues of some integral operator. Then we employ the Nyström method to discretize the integral operator and obtain a nonlinear matrix eigenvalue problem. The eigenvalues are computed using a multistep parallel spectral indicator method. Numerical examples demonstrate the high accuracy of the proposed method and can serve as the benchmarks. Our study provides a practical approach and can be extended to other scattering problems. This paper continues our previous study on the computation method for scattering poles of sound-soft obstacles.

Reconstructing the shape and material parameters of dissipative obstacles using an impedance model

In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine et al (2001 Asymptotic Anal. 26 257–83). We find that in certain regimes this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.

Bayesian model error method for the passive inverse scattering problem

This paper focuses on the passive inverse scattering problem, which uses passive measurements corresponding to randomly distributed incident sources to recover the shape of the sound-soft obstacle from a Bayesian perspective. Due to the unpredictability and randomness of incident sources, the classical Bayesian inversion framework may be unable to capture the likelihood involving the passive forward model for this inverse problem. We present the Bayesian model error method (BMEM), a novel passive imaging technique, to overcome this difficulty. The cross-correlations and the Helmholtz–Kirchhoff identity are specifically used to build an approximate active scattering model. This approximate model and the model error that it produces can be combined effectively by the suggested BMEM. The well-posedness of the posterior measure in the BMEM is proved. To further estimate the model error, an online scheme is utilized in conjunction with a preconditioned Crank–Nicolson Markov Chain Monte Carlo method to numerically approximate the posterior. Numerical experiments illustrate the effectiveness of the proposed method and also show that the online evaluation of model error can significantly improve reconstruction accuracy.

Computation of scattering poles using boundary integrals

Scattering resonances have important applications in mathematics, physics and engineering. They can be viewed as the poles of the meromorphic extension of the scattering operator. In this paper, we consider the computation of the scattering poles for sound soft obstacles. The scattering problem is formulated using boundary integrals, which is then discretized by the Nyström method. The discrete scattering poles are computed using the contour integral method. The proposed method is highly accurate, free of spurious modes, and can be extended to treat other obstacles, e.g., sound hard obstacles. Numerical examples are presented to validate the effectiveness and accuracy. The current paper is among the very few computational studies of scattering poles for obstacles.

A neural network warm-start approach for the inverse acoustic obstacle scattering problem

In this paper, we consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the L2 distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment.We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research.

Acoustic scattering and the exact Green function.

Acoustic waves are scattered by a bounded three-dimensional sound-soft obstacle. The scattered field can be given explicitly in terms of the exact Green function, defined as the solution of the scattering problem when the incident waves are generated by a point source. Methods for constructing the exact Green function are described, with emphasis on a recent paper by Rother and Hawkins [J. Acoust. Soc. Am. 149, 2179-2188 (2021)]. These methods include boundary integral equations and null-field equations.

Bayesian approach for limited-aperture inverse acoustic scattering with total variation prior

In this work, we apply the Bayesian approach for the acoustic scattering problem to reconstruct the shape of a sound-soft obstacle using the limited-aperture far-field measure data. A novel total variation prior scheme is developed for the obstacle shape parameterization. It is imposed on the Fourier coefficients of the parameterization not the parameterization itself. Using this prior, some less smooth objects can be reconstructed. We also investigate the well-posedness in the sense of the Hellinger distance, Wasserstein distance and Kullback–Leibler divergence. Extensive numerical tests are provided to illustrate the numerical performance.

An iterative scheme for imaging acoustic obstacle from phaseless total-field data

<p style='text-indent:20px;'>In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft or sound-hard obstacle from the modulus of the total-field collected on a measured curve for an incident point source. We propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.</p>

Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization

In this paper, we consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at a collection of receivers outside the object. The data is assumed to be generated by plane waves impinging on the unknown obstacle from multiple directions and at multiple frequencies. This inverse problem can be reformulated as an optimization problem: that of finding band-limited shape and impedance functions which minimize the L2 distance between the computed value of the scattered field at the receivers and the given measurement data. The optimization problem is highly non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate (since a large number of Helmholtz boundary value problems need to be solved at every iteration in the optimization loop). The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of an inhomogeneous object or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions of the object. The RLA is, in essence, a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton’s method at the new frequency. We demonstrate the effectiveness of this approach with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are generated by sound-hard or sound-soft objects, eliminating the need to know the precise boundary conditions appropriate for modeling the object under consideration. While the method is effective in obtaining high-quality reconstructions for many complicated geometries and impedance functions, a number of interesting open questions remain regarding the convergence behavior of the approach. We present numerical experiments that suggest underlying mechanisms of success and failure, pointing out areas where improvements could help lead to robust and automatic tools for the solution of inverse obstacle scattering problems.

Bayesian approach to inverse time-harmonic acoustic obstacle scattering with phaseless data generated by point source waves

This paper concerns the Bayesian approach to inverse acoustic scattering problems of inferring the position and shape of a sound-soft obstacle from phaseless far-field data generated by two-dimensional point source waves. Given the total number of obstacle parameters, the Markov chain Monte Carlo (MCMC) method is employed to reconstruct the boundary of the obstacle in a high-dimensional space, which usually leads to slow convergence and prohibitively high computational cost. We use the Gibbs sampling and preconditioned Crank–Nicolson (pCN) algorithm with random proposal variance to improve the convergence rate, and design an effective strategy for the surrogate model constructed by the generalized polynomial chaos (gPC) method to reduce the computational cost of MCMC. Numerical examples are provided to illustrate the effectiveness of the proposed method.

Reconstruction of an impenetrable obstacle in anisotropic inhomogeneous background

Abstract In this paper, we are interested in the problem of determining the shape of sound-soft or impedance obstacles in the acoustic wave scattering with an anisotropic inhomogeneous medium. The main theme is to extend the factorization method to this setting. Precisely, we can reconstruct the obstacle by the eigenvalues and eigenfunctions of the far-field operator. We also provide some numerical simulations in the paper.

Analysis of the Acoustic Waves Reflected by a Cluster of Small Holes in the Time-Domain and the Equivalent Mass Density

We study the time-domain acoustic scattering problem by a cluster of small holes (i.e., sound-soft obstacles). Based on the retarded boundary integral equation method, we derive the asymptotic expansion of the scattered field as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance between the holes, we show that the scattered field is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole and the causal signals (of these point-sources) can be computed by solving a retarded in time linear algebraic system. A rigorous justification of the asymptotic expansion and the unique solvability of the linear algebraic system is shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective acoustic medium (an equivalent mass density characterized by the capacitance of the holes) that generates, approximately, the same scattered field as the cluster of holes. Conversely, given a locally variable, smooth, and positive mass density, satisfying a certain subharmonicity condition, we can design a perforated material with holes, having appropriate capacitances, that generates approximately the same acoustic field as the acoustic medium modeled by the given mass density (and constant speed of propagation). Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions of the scattered fields via the finite element method.

Inverse scattering reconstruction of a three dimensional sound-soft axis-symmetric impenetrable object * *The work of JL was partially supported by the Funds for Creative Research Groups of NSFC (No. 11621101) and NSFC Grant No. 11871427.

In this work, we consider the problem of reconstructing the shape of a three dimensional impenetrable sound-soft axis-symmetric obstacle from measurements of the scattered field at multiple frequencies. This problem has important applications in locating and identifying obstacles with axial symmetry in general, such as, land mines. We obtain a uniqueness result based on a single measurement and propose a two-part framework for recovering the shape of the obstacle. In part 1, we introduce an algorithm to find the axis of symmetry of the obstacle by making use of the far field pattern. In part 2, we recover the shape of the obstacle by applying the recursive linearization algorithm (RLA) with multifrequency measurements of the scattered field. In the RLA, a sequence of inverse scattering problems using increasing single frequency measurements are solved. Each of those problems is ill-posed and nonlinear. The ill-posedness is treated by using a band-limited representation for the shape of the obstacle, while the nonlinearity is dealt with by applying the damped Gauss–Newton method. When using the RLA, a large number of forward scattering problems must be solved. Hence, it is paramount to have an efficient and accurate forward problem solver. For the forward problem, we apply separation of variables in the azimuthal coordinate and Fourier decompose the resulting problem, leaving us with a sequence of decoupled simpler forward scattering problems to solve. Numerical examples for the inverse problem are presented to show the feasibility of our two-part framework in different scenarios, particularly for objects with non-smooth boundaries.

An iterative ensemble Kalman method for an inverse scattering problem in acoustics

This paper studies an inverse problem of reconstructing a sound-soft obstacle from a homogeneous medium. We deal with it in the framework of statistical inversion and adopt an iterative ensemble Kalman algorithm to reconstruct the boundary. Some numerical examples show that the algorithm is effective and it can recover the shape of the boundary using one or several of the incident waves.

Bayesian approach to inverse time-harmonic acoustic scattering with phaseless far-field data

This paper is concerned with inverse acoustic scattering problem of inferring the position and shape of a sound-soft obstacle from phaseless far-field data. We propose the Bayesian approach to recover sound-soft disks, line cracks and kite-shaped obstacles through properly chosen incoming waves in two dimensions. Given the Gaussian prior measure, the well-posedness of the posterior measure in the Bayesian approach is discussed. The Markov Chain Monte Carlo (MCMC) method is adopted in the numerical approximation and the preconditioned Crank–Nicolson (pCN) algorithm with random proposal variance is utilized to improve the convergence rate. Numerical examples are provided to illustrate effectiveness of the proposed method.

Extended-Sampling-Bayesian Method for Limited Aperture Inverse Scattering Problems

Limited aperture inverse scattering problems arise in many important applications. In this paper, we propose a new method combining the extended sampling method (ESM) and the Bayesian approach for the inverse acoustic scattering problem to reconstruct the shape of a sound-soft obstacle using the limited aperture data. The problem is formulated as a statistical model using the Bayes formula. The well-posedness is proved in the sense of the Hellinger metric. A modified ESM is proposed to obtain the obstacle location, which is critical to the convergence of the MCMC algorithm. An extensive numerical study is presented to illustrate the performance of the method.

Inverse Fluid-solid Interaction Scattering Problem Using Phased and Phaseless Far Field Data

We consider the inverse fluid-solid interaction scattering of incident plane wave from the knowledge of the phased and phaseless far field patterns. For the phased data, one direct sampling method for location and shape reconstruction is proposed. Only inner product is involved in the computation, which makes it very simple and fast to be implemented. With the help of the factorization of the far field operator, we give a lower bound of the proposed indicator functional for the sampling points inside the elastic body. While for the sampling points outside, we show that the indicator functional decays like the Bessel function as the points go away from the boundaries of the elastic body. We also show that the proposed indicator functional continuously dependents on the far field patterns, which further implies that the novel sampling method is extremely stable with respect to data error. For the phaseless data, to overcome the translation invariance, we consider the scattering of point sources simultaneously. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Numerically, we introduce a phase retrieval algorithm to retrieve the phased data without the additional obstacle. The novel phase retrieval algorithm can also be combined with the sampling method for phased data. We also design two novel direct sampling methods using the phaseless data directly. Finally, some numerical simulations in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed numerical methods.

On controllability methods for the Helmholtz equation

When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is highly indefinite and thus notoriously difficult to solve, in fact increasingly so at higher frequency. The exact controllability approach (Bristeau et al., 1994) instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period, the controllability method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. The original energy functional used to penalize the departure from periodicity is strictly convex only for sound-soft scattering problems. To extend the controllability approach to general boundary-value problems governed by the Helmholtz equation, new penalty functionals are proposed, which are numerically efficient. Numerical experiments for wave scattering from sound-soft and sound-hard obstacles, inclusions, but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods.

Inverse Acoustic Scattering with Phaseless Far Field Data: Uniqueness, Phase Retrieval, and Direct Sampling Methods

The well-known translation invariance property of the phaseless far field patterns with incident plane waves make it is impossible to reconstruct the location of the underlying scatterers even for multiple incident directions and frequencies. To overcome this difficulty, we take the superpositions of plane waves and point sources as the incident fields. The point sources have fixed source location with at most three different scattering strengths. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Of possibly even more interest are the phase retrieval technique and direct sampling methods, without using the reference obstacle. With the help of the point sources, we propose a simple and stable phase retrieval algorithm and design some novel direct sampling methods using the phaseless data directly. The novel phase retrieval algorithm can then be combined with the classical sampling methods for scatterer reconstruction. Finally, we present a variant of numerical examples in two dimensions to verify the effectiveness and robustness of the proposed algorithms.

A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft obstacle from the modulus of the far-field data for a single incident plane wave. By adding a reference ball artificially to the inverse scattering system, we propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. The reference ball technique causes few extra computational costs, but breaks the translation invariance and brings information about the location of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.