Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients

In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the phased and phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and phaseless cases. An optimal strategy is suggested to attain the infimum of the condition numbers of the phaseless reconstruction, which may provide an important guidance for efficient phaseless measurements in practical applications. To the best of our knowledge, the stability analysis in terms of condition numbers are new for the phased and phaseless inverse scattering problems, and are very important to help us understand the degree of ill-posedness of these inverse problems. Numerical experiments are provided to illustrate the theoretical asymptotic behavior, as well as the effectiveness and robustness of the phaseless reconstruction algorithm

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.Comment: 20 pages, 13 figure

An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper concerns an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to establish the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nystr\"{o}m-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is also proposed for the phaseless inverse problem. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed methods.Comment: arXiv admin note: text overlap with arXiv:1811.1257

Uniqueness of a 3-D coefficient inverse scattering problem without the phase information

We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells

Uniqueness in inverse acoustic scattering with phaseless near-field measurements

This paper is devoted to the uniqueness of inverse acoustic scattering problems with the modulus of near-field data. By utilizing the superpositions of point sources as the incident waves, we rigorously prove that the phaseless near-fields collected on an admissible surface can uniquely determine the location and shape of the obstacle as well as its boundary condition and the refractive index of a medium inclusion, respectively. We also establish the uniqueness in determining a locally rough surface from the phaseless near-field data due to superpositions of point sources. These are novel uniqueness results in inverse scattering with phaseless near-field data.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1812.0329

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

Machine learning based data retrieval for inverse scattering problems with incomplete data

We are concerned with the inverse scattering problems associated with incomplete measurement data. It is a challenging topic of increasing importance in many practical applications. Based on a prototypical working model, we propose a machine learning based inverse scattering scheme, which integrates a CNN (convolution neural network) for the data retrieval. The proposed method can effectively cope with the reconstruction under limited-aperture and/or phaseless far-field data. Numerical experiments verify the promising features of our new scheme.Comment: The authors withdrew the previous three versions because more work was needed to furnish before its appearance as a formal paper. The current version is fine. All helpers agree with current versio

Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency

This paper is concerned with uniqueness in inverse acoustic scattering with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern generated by one plane wave as the incident field. Based on our previous work (J. Comput. Phys. 345 (2017), 58-73), the translation invariance property of the phaseless far-field pattern can be broken by using infinitely many sets of superpositions of two plane waves as the incident fields at a fixed frequency. In this paper, we prove that the obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency under the condition that the obstacle is a priori known to be a sound-soft or non-absorbing impedance obstacle and the index of refraction $n$ of the inhomogeneous medium is real-valued and satisfies that either $n-1\ge c_1$ or $n-1\le-c_1$ in the support of $n-1$ for some positive constant $c_1$. To the best of our knowledge, this is the first uniqueness result in inverse scattering with phaseless far-field data. Our proofs are based essentially on the limit of the normalized eigenvalues of the far-field operators which is also established in this paper by using a factorization of the far-field operators

Reconstruction of acoustic sources from multi-frequency phaseless far-field data

We consider the inverse source problem of determining an acoustic source from multi-frequency phaseless far-field data. By supplementing some reference point sources to the inverse source model, we develop a novel strategy for recovering the phase information of far-field data. This reference source technique leads to an easy-to-implement phase retrieval formula. Mathematically, the stability of the phase retrieval approach is rigorously justified. Then we employ the Fourier method to deal with the multi-frequency inverse source problem with recovered phase information. Finally, some two and three dimensional numerical results are presented to demonstrate the viability and effectiveness of the proposed method

Bayesian Approach to Inverse Time-harmonic Acoustic Scattering from Sound-soft Obstacles with Phaseless Data

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 point source waves. To improve the convergence rate, we use the Gibbs sampling and preconditioned Crank-Nicolson (pCN) algorithm with random proposal variance to implement the Markov chain Monte Carlo (MCMC) method. This usually leads to heavy computational cost, since the unknown obstacle is parameterized in high dimensions. To overcome this challenge, we examine a surrogate model constructed by the generalized polynomial chaos (gPC) method to reduce the computational cost. Numerical examples are provided to illustrate the effectiveness of the proposed method