Special Session 127: Recent Advances in Inverse Problems, Imaging, and Their Applications

Uniqueness of an inverse cavity scattering problem for the time-harmonic biharmonic wave equation

Heping Dong School of Mathematics, Jilin University Peoples Rep of China Co-Author(s):    Peijun Li

Abstract: This talk addresses an inverse cavity scattering problem associated with the biharmonic wave equation in two dimensions. The objective is to determine the domain or shape of the cavity. The Green`s representations are demonstrated for the solution to the boundary value problem, and the one-to-one correspondence is confirmed between the Helmholtz component of biharmonic waves and the resulting far-field patterns. Two mixed reciprocity relations are deduced, linking the scattered field generated by plane waves to the far-field pattern produced by various types of point sources. Furthermore, the symmetry relations are explored for the scattered fields generated by point sources. Finally, we present two uniqueness results for the inverse problem by utilizing both far-field patterns and phaseless near-field data.

Analysis of Subwavelength Resonances in High Contrast Elastic Media

Yiixian Gao Northeast Normal University Peoples Rep of China Co-Author(s):

Abstract: This talk is devoted to the mathematical study of the wave scattering by a hard elastic obstacle embedded in a soft elastic body in three-dimensional space. We present the asymptotic expansions of the subwavelength resonant frequencies in the low-frequency region by an explicit characterization. We give the representation of the scattered field in the interior domain, whose enhancement coefficients are determined by the imaginary parts of the resonance frequencies. Moreover, we also establish the transversal and longitudinal far-field patterns for the scattered field the exterior domain.

Reduced order model approach for imaging with waves

Josselin Garnier Ecole polytechnique France Co-Author(s):    Liliana Borcea Alex Mamonov Jorn Zimmerling

Abstract: We consider the inverse problem for the scalar wave equation. Sensors probe the unknown medium to be imaged with a pulse and measure the backscattered waves. The objective is to estimate the velocity map from the array response matrix of the sensors. Under such circumstances, conventional Full Waveform Inversion (FWI) can be carried out by nonlinear least-squares data fitting. It turns out that the FWI misfit function is high-dimensional and non-convex and it has many local minima. A novel approach to FWI based on a data-driven reduced order model (ROM) of the wave equation operator is introduced and it is shown that the minimization of ROM misfit function performs much better.

Time-domain and frequency-domain methods to inverse moving source problems

Guanghui Hu Nankai University Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with uniqueness, stability and algorithms for inverse moving point source problems modeled by the acoustic wave equation. The purpose is to recover the orbit of a moving point source from the dynamical data recorded at a finite number of observation points. In the time domain, we derive an ordinary differential equation for the distance function between an observation point and the moving target. Solving such ODEs at four observation points yields the orbit function of the moving source. The frequency-domain method is to Fourier-transform the time-dependent source problem of the wave equation into an equivalent source problem of the Helmholtz equation with multi-frequency near-field data. This turns out to be a special wavenumber-dependent inverse source problem in the time-harmonic regime. We shall discuss the concept of non-observation directions and a non-iterative approach for imaging the orbit function. A comparision of the time-domain and frequency-domain method will be remarked at the end of the talk.

Functional normalizing flows for statistical inverse problems of partial differential equations

Junxiong Jia Xi`an Jiaotong University Peoples Rep of China Co-Author(s):    Yang Zhao and Deyu Meng

Abstract: To assess the uncertainties inherent in the inverse problems associated with partial differential equations, we employ Bayes` theorem to frame these as statistical inference challenges. Recently, well justified infinite-dimensional Bayesian analysis methods have been developed to construct dimension-independent algorithms. However, it is difficult to sample directly from the posterior of the Bayes` formula. To address this issue, a normalizing flows based infinite-dimensional variational inference (NF-iVI) has been proposed. Specifically, by introducing some well defined transformations, the prior of the Bayes` formula will be transformed into some complex measures. We will use the post-transformed measures to construct posterior approximations. To avoid the mutually singular obstacle that occurred in the infinite-dimensional variational inference approach, we proposed a rigorous theory of the transformations so that the post-transformed measure will be equivalent to the pre-transformed measure. In addition, we also introduce the conditional normalizing flows based infinite-dimensional variational inference (CNF-iVI), which can mitigate the computational cost of NF-iVI. The proposed algorithms are applied to two inverse problems governed by the simple smooth equation and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of our algorithms, and verify the mesh-independent property.

Inverse random potential scattering for the polyharmonic wave equation using far-field patterns

Jianliang Li Hunan Normal University Peoples Rep of China Co-Author(s):    Peijun Li, Xu Wang, Guanlin Yang

Abstract: This talk addresses the inverse scattering problem of a random potential associated with the polyharmonic wave equation in two and three dimensions. The random potential is represented as a centered complex-valued generalized microlocally isotropic Gaussian random field, where its covariance and relation operators are characterized as conventional pseudo-differential operators. Regarding the direct scattering problem, the well-posedness is established in the distribution sense for sufficiently large wavenumbers through analysis of the corresponding Lippmann-Schwinger integral equation. Furthermore, in the context of the inverse scattering problem, the uniqueness is attained in recovering the microlocal strengths of both the covariance and relation operators of the random potential. Notably, this is accomplished with only a single realization of the backscattering far-field patterns averaged over the high-frequency band.

Direct sampling methods for inverse source problems

Xiaodong Liu Academy of Mathematics and Systems Science of Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Xiaodong Liu

Abstract: This talk is dedicated to a short review of the direct sampling methods for inverse source problems. In particular, we show that the recently developed direct sampling method is able to give a high resolution imaging for the source support and to produce an acceptable reconstruction of the source function.

A high-order fast sweeping method for eikonal and transport equations in attenuating media

Wangtao Lu Zhejiang University Peoples Rep of China Co-Author(s):    Gang Bao, Tianlu Chen, Wangtao Lu and Jianliang Qian

Abstract: Eikonal and transport equations arise from using the geometrical-optics ansatz in solving wave equations in the high-frequency regime. In attenuating media, the unknowns are complex-valued so that the real and imaginary parts are coupled with each other. In this talk, we present an effective fast sweeping solver for solving the two equations. Based on a specially designed numerical Hamiltonian, we develop a fast Gauss-Seidel iterative scheme, and establish its convergence theory. Numerical experiments are carried out to demonstrate the effectiveness of the new scheme.

A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems

Dong Wang The Chinese University of Hong Kong, Shenzhen & Shenzhen International Center for Industrial and Applied Mathematics Peoples Rep of China Co-Author(s):    Dong Wang, Chinese University of Hong Kong, Shenzhen

Abstract: In this talk, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM.

Inverse random potential scattering for stochastic polyharmonic wave equations

Xu Wang Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Peijun Li and Guanlin Yang

Abstract: In this talk, we mainly discuss the uniqueness of the inverse scattering problem for the random potential involved in stochastic polyharmonic wave equations. The random potential is assumed to be an isotropic generalized Gaussian random field. With limited measurements, we show the unique determination of the correlation strength of the random potential through a single realization of the scattered wave field averaged over the frequency band.

Near-field inverse obstacle scattering by flexural waves: method of transformed field expansion

Yuliang Wang Beijing Normal University Peoples Rep of China Co-Author(s):    Peijun Li, Yuliang Wang

Abstract: In this talk, we investigate the inverse scattering problem of an obstacle embedded in a thin plate using near-field measurements of flexural waves. The forward scattering problem is reduced to a coupled system of boundary value problems for the propagating and evanescent waves. Assuming the obstacle is a small perturbation of a circle, we employ the method of transformed field expansions to express the solution as a power series, obtaining closed-form expressions for the zeroth and first-order terms. These expressions are then used to derive an approximate reconstruction formula for the inverse scattering problem. We explore different types of incident fields, some of which lead to simplified and more efficient reconstruction methods. Numerical experiments demonstrate the effectiveness and efficiency of the proposed approach.

Unsupervised diffusion approach with null space learning for cloud removal in remote sensing images

Liwei Xu University of Electronic Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: Clouds are ubiquitous in remote sensing images. Most of the existing methods for cloud removal are limited to either implementing on multi spectral images or exploiting supervised learning technique. In this paper, we propose an unsupervised diffusion approach by deploying the null space learning. The proposed approach is built upon two trained denoising diffusion probabilistic models by diverse remote sensing datasets so as to tackle the mixture of data from different sources. The simplified degradation and self-adaptive generalized inverse matrices are devised for the null space decomposition. For the diffusion model with null space decomposition, we derive its continuous reverse-time stochastic differential equation (SDE), which is theoretically proven to be variance preserving. We further derive the explicit formula for the expectation of the reverse-time SDE, which is conducive to algorithm improvement. As a byproduct, the proposed approach can also be applicable to the transparency separation. Numerical experiments on some remote sensing images demonstrate that the proposed approach outperforms some state-of-the-art unsupervised, even supervised, cloud removal methods.

Inverse random potential scattering for stochastic polyharmonic wave equations

Guanlin Yang Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Guanlin Yang

Abstract: In this talk, we mainly discuss the uniqueness of the inverse scattering problem for the random potential involved in stochastic polyharmonic wave equations. The random potential is assumed to be an isotropic generalized Gaussian random field. With limited measurements, we show the unique determination of the correlation strength of the random potential through a single realization of the scattered wave field averaged over the frequency band.

DtN-FEM for thermoelastic scattering problem

Tao Yin Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: This talk will present our recent works on the wellposedness analysis and numerical schemes for solving the thermo/poro-elastic scattering problems. Based on the Helmholtz decomposition, the vector coupled governing equations of thermoelastic wave are decomposed into three Helmholtz equations of scalar potentials with different wavenumbers. Then the Dirichlet-to-Neumann (DtN) map and the corresponding transparent boundary condition are constructed by using Fourier series expansions of the scalar potentials. The well-posedness results are established for the variational problem and its modification due to the truncation of the DtN map. A priori and a posteriori error estimates, including both the effects of the finite element approximation and truncation of the DtN operator, are derived. Numerical experiments are presented to validate the theoretical results.

Convergence of the TBC/PML method for the biharmonic wave scattering problem in periodic structures

Xiaokai Yuan Jilin University Peoples Rep of China Co-Author(s):    Gang Bao and Peijun Li

Abstract: This talk investigates the scattering of biharmonic waves by a one-dimensional periodic array of cavities embedded in an infinite elastic thin plate. The transparent boundary conditions are introduced to formulate the problem from an unbounded domain to a bounded one. The well-posedness of the associated variational problem is demonstrated utilizing the Fredholm alternative theorem. The perfectly matched layer (PML) method is employed to reformulate the original scattering problem, transforming it from an unbounded domain to a bounded one. The transparent boundary conditions for the PML problem are deduced, and the well-posedness of its variational problem is established. Moreover, exponential convergence is achieved between the solution of the PML problem and that of the original scattering problem.

Iterative regularized contrast source inversion type methods for the inverse medium scattering problem

Haiwen Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Qiao Hu and Bo Zhang

Abstract: This talk is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data. The contrast source inversion (CSI) methods are the well-known algorithms for such kind of inverse scattering problem, which are very fast and efficient. Recently, we propose two iterative regularized CSI-type methods. Our methods have very low computational complexity. Moreover, we prove the global convergence of the proposed methods. Numerical experiments show that our methods are very robust and have faster convergence rates than the original CSI-type methods.

INVERSE PROBLEMS FOR NON-LINEAR FRACTIONAL MAGNETIC SCHRODINGER EQUATIONS

Ting Zhou Zhejiang University Peoples Rep of China Co-Author(s):    Ting Zhou and Ru-Yu Lai

Abstract: This talk focus on the forward problem and inverse problem for the fractional magnetic Schrodinger equation with nonlinear electric potential. We first investigate the maximum principle for the linearized equation and apply it to show that the problem is well-posed under suitable assumptions on the exterior data. Moreover, we explore uniqueness of recovery of both magnetic and electric potentials.