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.
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…
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 generalize…
Phaseless Imaging by Reverse Time Migration: Acoustic Waves
Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Department of Computational and Applied Mathematics, Rice University（莱斯大学计算与应用数学系）
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 correspond…
This paper is concerned with the inverse acoustic scattering problem with phaseless total-field data at a fixed frequency. An approximate factorization method is developed to numerically reconstruct both the location and shape of the unknown scatterer from the phaseless total-field data generated by incident plane waves at a fixed frequency and measured on the circle $\partial B_R$ with a sufficiently large radius $R$. The theoretical analysis of our method is based on the asymptotic property in the operator norm from $H^{1/2}(\mathbb{S}^1)$ to $H^{-1/2}(\mathbb{S}^1)$ of the phaseless total-f…
This paper is concerned with reconstructing an acoustic obstacle and its excitation sources from the phaseless near-field measurements. By supplementing some artificial sources to the inverse scattering system, this co-inversion problem can be decoupled into two inverse problems: an inverse obstacle scattering problem and an inverse source problem, and the corresponding uniqueness can be established. This novel decoupling technique requires some extra data but brings in several salient benefits. First, our method is fast and easy to implement. Second, the boundary condition of the obstacle is …
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；NCMIS（数学与应用数学国家重点实验室）；LSEC（计算数学与科学工程计算研究所）
In optical diffraction tomography (ODT), the three-dimensional scattering potential of a microscopic object rotating around its center is recovered by a series of illuminations with coherent light. Reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane, whereas often only intensities, i.e., phaseless measurements, are available in practice. We propose a new reconstruction approach for ODT with unknown phase information based on three key ingredients. First, the light propagation is modeled using Born's approximation …
An iterative approach to monochromatic phaseless inverse scattering
Max-Planck Institute for Solar Systems Research（马克斯·普朗克太阳系研究所）；Institute for Numerical and Applied Mathematics, University of Göttingen（哥廷根大学数值与应用数学研究所）；CMAP, Ecole Polytechnique, CNRS, Université Paris-Saclay（巴黎萨克雷大学，法国国家科学研究中心，综合理工学院计算数学与应用中心）；IEPT RAS（俄罗斯科学院普罗索金动力学与控制问题研究所）
A novel sampling method for reconstruction of a moving point acoustic source in $\mathbb{R}^3$
School of Mathematical Sciences, Sichuan Normal University（四川师范大学数学科学学院）；Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学与交叉科学研究院）
Multi-frequency iterative methods for the inverse medium scattering problems in elasticity
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Department of Computing & Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学系）
In this paper an even higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-acc…
Optimal transport theory has recently reemerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology -- the reconstruction of the linear density field from low redshifts, in particular the recovery of the Baryonic Acoustic Oscillation (BAO) scale. We demonstrate our algorithm's accuracy by retrieving the BAO scale in noise-less cosmological simulations that are dedicated to cancel cosmic variance; we find uncertainties…
Most methods tackling the phase retrieval problem of magnitude-only antenna measurements suffer from unrealistic sampling requirements, from unfeasible computational complexities, and, most severely, from the lacking reliability of nonlinear and nonconvex formulations. As an alternative, we propose a partially coherent (PC) multi-probe measurement technique and an associated linear reconstruction method which mitigate all these issues. Hence, reliable and accurate phase retrieval can be achieved in near-field far-field transformations (NFFFTs). In particular, we resolve the issues related to o…
This paper is concerned with the numerical simulation of three dimensional time-dependent inverse source problems of acoustic waves. The reconstructions of both multiple stationary point sources and a moving point source are considered. The modified method of fundamental solutions (MMFS), which expands the solution utilizing the time convolution of the Green's function and the signal function, is proposed to solve the problem. For the reconstruction of a moving point source, moreover, the MMFS is simplified as a simple sampling method at each time step. Numerical experiments are provided to sh…
Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging technique that holds great promise for biomedical imaging. PACT is a hybrid imaging method that can exploit the strong endogenous contrast of optical methods along with the high spatial resolution of ultrasound methods. In its canonical form that is addressed in this article, PACT seeks to estimate the photoacoustically-induced initial pressure distribution within the object. Image reconstruction methods are employed to solve the acoustic inverse problem associated with the image formation p…
