Special Session 43: Recent Advances in Inverse Problems, Imaging, and Their Applications

Data driven reduced order modeling approach to inverse scattering

Liliana Borcea Columbia University USA Co-Author(s):    Josselin Garnier, Alexander Mamonov, Jorn Zimmerling

Abstract: Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. The traditional formulation, called full waveform inversion (FWI), estimates unknown coefficients in wave equations, that model the medium, via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to learn a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves.

Reconstruction Algorithms for Multi-Slice Ptychography Based on Chain-Structured Modeling

Huibin Chang Tianjin Normal University Peoples Rep of China Co-Author(s):

Abstract: Ptychography has emerged as a premier lensless imaging technique, offering sub-Angstrom resolution and exceptional phase contrast for characterizing nanomaterials. However, traditional 2D models rely on the thin-slice approximation, which fails to account for multiple scattering in thick or strongly scattering specimens. Existing multi-slice ptychography (msPtycho) algorithms, such as 3PIE, are often derived heuristically, lack rigorous convergence guarantees, and suffer from slow convergence or heavy computational burdens. To address these challenges, this talk introduces a novel chain-structured modeling approach that reduces complex multi-linear couplings into a sequence of bilinear relations. Based on this framework, we present the AM2SP algorithm, providing closed-form solutions and proven global convergence to stationary points. By incorporating forward-propagating correction and Nesterov-like extrapolation, the enhanced AM2SP-FX algorithm significantly suppresses error accumulation and accelerates reconstruction speed by over an order of magnitude compared to traditional baselines. Furthermore, we explore the integration of Implicit Neural Representation (INR) to maintain high-fidelity reconstruction quality even under large-step scanning geometries with low overlap ratios. This synergy between rigorous physical modeling and continuous neural representation offers a robust, scalable solution for high-resolution 3D imaging.

Uniqueness of an inverse cavity problem for the biharmonic equation

Heping Dong Jilin University Peoples Rep of China Co-Author(s):

Abstract: This talk concerns cavity problems for the biharmonic equation in an unbounded Kirchhoff--Love plate. By applying a concentrated point force inside the plate and using two sets of boundary measurements, we address the inverse problem of determining an unknown cavity. A key component of the analysis is the construction of a Dirichlet-to-Neumann (DtN) map, which reduces the unbounded domain problem to an equivalent formulation on a bounded domain. This reduction enables a variational analysis and establishes the well-posedness of the direct problem and the uniqueness in recovering both the location and shape of the cavity by using of a unique continuation principle.

Subwavelength resonances and bandgaps in high contrast elastic system

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

Abstract: This talk develops a rigorous mathematical framework for subwavelength resonances and bandgaps in high-contrast elastic systems. Using variational methods, layer potential techniques, and Floquet-Bloch theory, we study 2D and 3D elastic wave scattering, establish a solid mathematical foundation for phononic crystal band structures, and provide reliable theoretical support for wave control and elastic metamaterial design.

Effective sound speed estimation in ultrasound imaging

Josselin Garnier Ecole polytechnique France Co-Author(s):    Laure Giovangigli, Quentin Goepfert, Pierre Millien

Abstract: In this talk, we present a mathematical model and analysis for a new method to estimate the sound speed in ultrasound imaging. We first perform a detailed analysis of the point-spread function of an imaging system in the presence of a mismatch between the true sound speed in the medium and the speed used in the reverse-time imaging function. This analysis leads to an estimator of the sound speed in the presence of a point-like reflector (a guide star). In a second part, we consider a random multiscale medium (modeling biological tissue for instance) and use stochastic homogenization techniques to derive a representation formula for the scattered field. We then show that statistical moments of the imaging functional can be recovered from data corresponding to a single realization of the medium. We demonstrate that the point-spread function can be extracted directly from speckle patterns, making it possible to estimate the effective sound speed even in the absence of a point-like reflector.

Shape optimization driven regularization methods for bioluminescence tomography

Rongfang Gong Nanjing University of Aeronautics and Astronautics Peoples Rep of China Co-Author(s):    Wei Gong, Shengfeng Zhu, Qianqian Wu, Ziyi Zhang

Abstract: In this talk, we investigate the inverse source problem arising in bioluminescence tomography, the objective of which is to reconstruct both the support and the intensity of an internal light source from boundary measurements governed by an elliptic model. A shape optimization framework is developed in which the source intensity and its support are decoupled through first-order optimality conditions. To enhance the stability of the reconstruction, we incorporate a parameter-dependent coupled complex boundary method together with perimeter and volume regularizations. Source support is represented by a level set function, allowing the algorithm to naturally accommodate topological changes and recover multiple, closely spaced, or nested source regions. Theoretical justifications for the proposed formulation and regularization strategy are established, and extensive numerical experiments are performed to assess the reconstruction accuracy for both noise-free and noisy data. The results demonstrate that our method achieves robust and accurate recovery of source geometry and intensity, exhibits clear advantages over existing approaches.

Nonlinear Transformation Based Infinite-Dimensional Variational Inference for Statistical Inverse Problems

Junxiong Jia Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: Inverse problems for PDEs are common in scientific disciplines and can be formulated as statistical inference problems via Bayes theorem. For large-scale problems, developing discretization-invariant algorithms is crucial, achievable by formulating methods in infinite-dimensional spaces. Restricting the variational family to the pushforward of a prior measure`s nonlinear transformation yields various variational inference methods. Overcoming singularity issues in infinite-dimensional function spaces, we develop two methods: infinite-dimensional Stein variational gradient descent (iSVGD) and functional normalizing flows (FNF). The transformations in both iSVGD and FNF involve a sequence of identity operator perturbations. In iSVGD, perturbation mappings are in a reproducing kernel Hilbert space, while in FNF, they are constructed with designed neural network architectures. We apply these algorithms to an inverse problem of the steady-state Darcy flow equation. Numerical results validate the theoretical analysis, show the algorithms` efficiency, and confirm their discretization-invariant properties.

Selective focusing of multiple particles in a layered medium

Jun Lai Zhejiang University Peoples Rep of China Co-Author(s):    Jinrui Zhang

Abstract: Inverse scattering in layered media has a wide range of applications, including geophysical exploration, submarine detection, and medical imaging. In this talk, we develop a selective focusing method for identifying multiple unknown buried scatterers in a layered medium. The method is derived through asymptotic analysis of the time reversal operator, which is challenging due to the absence of an explicit form for the layered Green`s function and the limited aperture measurements. We demonstrate that each small sound-soft and sound-hard particle gives rise to one and three significant eigenvalues, respectively, and the corresponding eigenfunction generates an incident wave focusing selectively on each unknown particle. Finally, we employ the time reversal method as an initial indicator and propose an effective Bayesian inversion scheme to reconstruct multiple buried extended scatterers for enhanced resolution. Numerical experiments are provided to demonstrate the efficiency.

On the identification of external forces for two-layer quasi-geostrophic model from partial observation data

Jijun Liu Southeast University Peoples Rep of China Co-Author(s):

Abstract: The two-layer quasi-geostropic model can be considered as a simplified version of 3-dimensional Navier-Stokes equation governing the evolution process of ocean flow in stratified homogeneous medium. The potential vorticity and the stream functions in two adjoint layers are coupled together in a nonlinear PDE system, representing the dynamical evolution by external forces and the interactions between two layers. Due to the large scale of the spatial domain for ocean flow, the external forces cannot be observed directly. We consider an inverse problem for the recovery of external forces which are the main factors motivating the ocean flow evolution, with the stream functions measured only in part of the spatial domain as inversion input. Based on the equivalent representation of the solution to PDE systems, this nonlinear ill-posed problem is decomposed into two problems: one is a linear ill-posed problem essentially based on the extension of the solutions of linear elliptic system, the other is a nonlinear ill-posed problem from the numerical differentiations. We establish a reconstruction scheme by solving these two ill-posed problems with rigorous mathematical analysis, including the solvability of the regularizing system, the optimal choice strategy for regularizing parameters, and the error estimates on the recovered solution. The relation between the noise level of inversion input data, the resolution accuracy of the unknown sources and the reconstruction errors is quantitatively characterized by matrix decomposition and Fourier analysis techniques. Numerical implementations are presented to show the validity of our proposed scheme.

Inverse scattering problems from the multi-frequency backscattering data

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

Abstract: Inverse scattering from backscattering data is a foundational and impactful research area at the intersection of applied mathematics, computational physics, and engineering, with profound implications for critical real-world applications such as non-destructive testing, medical imaging, radar systems, and remote sensing. This research direction focuses on retrieving the geometric and physical properties of unknown scatterers (e.g., obstacles, inhomogeneous media) using only backscattered wave measurements-where the source and receiver are co-located or aligned in a backscattering configuration-a setup that is widely adopted in practical scenarios due to its simplicity and operational efficiency. In this talk, we introduce some uniqueness and numerical algorithms on this topic. In particular, our numerical algorithms avoid computing the direct problems.

Numerical algorithms for high-frequency wave propagation problems and related Hamilton-Jacobian equations

Wangtao Lu Zhejiang University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present fast algorithms for computing the highly oscillatory solutions in high-frequency wave propagation problems. The method is based on asymptotic ansatz which gives rise to unknowns independent of the frequency, governed by coupled Hamilton-Jacobian equations usually. I will show some theoretical and numerical results regarding solving such hyperbolic equations.

Efficient Numerical Methods for an Inverse Source Problem of the Wave Equation via Droplet-Induced Asymptotics

Haibing Wang Southeast Univerisity Peoples Rep of China Co-Author(s):    Shutong Hou and Mourad Sini

Abstract: In this talk, we show two novel approaches for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. First, we derive an asymptotic expansion of the wave field after the droplet injection, using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. Then two novel numerical methods for reconstructing the source via the expansion and mollification-based numerical differentiation. Our methods require only single-point measurements, overcoming traditional spatial data limitations. Several numerical experiments are presented to demonstrate the performance of the proposed methods.

Research on nonradiating sources of Maxwell`s equations

Jue Wang Hangzhou Normal University Peoples Rep of China Co-Author(s):

Abstract: This talk presents a thorough investigation into nonradiating sources of Maxwell equations. Various characterizations are developed to clarify the properties of nonradiating sources, considering their varying degrees of regularity. Furthermore, the characterizations are examined on far-field patterns and near-field data of the electric field, along with the null spaces of integral operators. The study includes the explicit construction of several illustrative examples to demonstrate the presence of nonradiating sources in different null spaces.

Multi-Frequency Unsupervised Deep Learning for Precise Inverse Acoustic Scattering

Yuliang Wang Wenzhou-Kean University Peoples Rep of China Co-Author(s):

Abstract: This talk presents a robust unsupervised deep learning approach for reconstructing the location and shape of impenetrable sound-soft obstacles in two dimensions from scattered field data. The method formulates the inverse acoustic scattering problem as a PDE-constrained shape optimization, where the obstacle boundary is parameterized by a neural network, and the forward problem is solved efficiently using the Discrete Source Method. Reconstruction accuracy is enhanced through a multi-frequency continuation strategy, in which low-frequency reconstructions are progressively refined using higher-frequency data. Moreover, targeting phaseless far-field data, we propose an imaging algorithm for obstacle localization, which is coupled with deep learning approach to thereby reconstruct the obstacle`s shape with high precision. The approach leverages automatic differentiation to compute gradients, avoiding the need for adjoint solvers, and integrates multi-angle measurements for improved stability. Numerical experiments demonstrate accurate recovery of complex geometries, including sharp corners and multiple obstacles, even under sparse sampling and severe noise.

Direct and inverse problems in thermoelasticity

Tao Yin Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: This talk will present a brief summary of our recent works on the direct and inverse thermoelastic wave scattering problems, including the spectral analysis of regularized boundary integral equation methods and numerical experiments for the problem of thermoelastic scattering by an open-arc, and some stability estimates for the inverse source problems in thermoelasticity.

Convergence of the DtR methods for the wave scattering by periodic structures

Xiaokai Yuan Jilin University Peoples Rep of China Co-Author(s):

Abstract: This talk presents a rigorous analysis of the Dirichlet-to-Robins (DtR) method for acoustic and elastic wave scattering by periodic structure. The variational problem is formulated by employing a DtR-based transparent boundary condition. The analysis establishes that the truncated problem is well-posed provided the DtR operator incorporates all propagating modes. Furthermore, through both a posteriori and a priori error estimates, it is proved that the solution obtained with the truncated DtR operator converges exponentially to the exact solution as the truncation number N increases.

Recovery of weakly anisotropic elastic parameters

Jian Zhai Fudan University Peoples Rep of China Co-Author(s):

Abstract: We consider the recovery of an anisotropic perturbation of an isotropic elastic body. When the reference isotropic body is homogeneous we show that the anisotropic perturbation can be fully determined by single-scattered waves. When the reference model is heterogeneous, the problem can be reduced to certain tensor tomography problems for which related results will also be discussed.

Uniform far-field asymptotics for scattering by unbounded rough surfaces and application to the inverse problem

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

Abstract: This talk is concerned with direct and inverse scattering by an unbounded rough surface with the Dirichlet boundary condition in two dimensions. We study the asymptotics of the solution to the considered scattering problem with the boundary data in a weighted space of continuous functions. Based on this, we develop a direct imaging method for recovering the unbounded rough surface from the scattered-field data generated by incident point sources. Numerical experiments are carried out to demonstrate the feasibility and robustness of our algorithms.

Inverse Problems of Output Feedback Controller via Contraction Metrics

Lei Zhang Zhejiang University of Technology Peoples Rep of China Co-Author(s):

Abstract: This talk presents an output feedback robust model predictive control (MPC) method for stabilizing nonlinear systems with additive disturbances and noisy output measurements. A robustly stable observer updates the estimation error set magnitude online, which is then used in the controller design for a predictive control inverse problem. By leveraging the sublevel set defined by the Riemannian distance, a set-valued system is constructed around the predicted trajectory. Reachability analysis parameterizes this set-valued system to obtain tube dynamics, forming the basis for robust constraint conditions. The proposed framework removes the need for explicit geodesics, avoiding online synthesis of auxiliary controllers. Additionally, two exponentially convergent observers are developed through contraction analysis, and a method for improving output feedback via online set-valued estimation is introduced. Numerical simulations show the effectiveness of the proposed approach in stabilizing the system while satisfying state and control input constraints.

Generative models for solving linear and nonlinear inverse problems

Xiaoqun Zhang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Xiang Cao, Qiaoqiao Ding

Abstract: Diffusion models have emerged as powerful tools for solving nonlinear inverse problems, offering high-quality reconstructions through conditional reverse sampling. However, challenges persist in handling partial differential equation (PDE)-constrained inverse problems, including computational inefficiency, discretization errors, and the inherent ill-posedness of nonlinear systems such as travel-time tomography and ultrasound computed tomography (USCT). In this talk, we present two synergistic frameworks that address these challenges by integrating diffusion priors with PDE-aware numerical and learned solvers. First, we propose a subspace diffusion approach for PDE-based travel-time tomography, introducing (1) a plug-and-play posterior sampling process that leverages adjoint-state equations for gradient updates and (2) a coarse-to-fine subspace acceleration technique to reduce sampling time while preserving reconstruction quality. Second, we develop Diff-ANO (Diffusion-based Models with Adjoint Neural Operators), a unified framework for Helmholtz equation-constrained USCT that combines (1) a conditional consistency model for few-step, measurement-conditional sampling and (2) neural operator surrogates to replace traditional PDE solvers, enabling memory-efficient adjoint gradient computation via our Batch-based Convergent Born Series (BCBS) strategy. Our experiments demonstrate significant improvements in both reconstruction accuracy and computational efficiency across sparse-view and partial-view measurement scenarios. The proposed methods bridge the gap between generative priors and PDE constraints, offering scalable solutions for nonlinear inverse problems in imaging.

Neural network yields regularization for ill-posed inverse problems

Ye Zhang Shenzhen MSU-BIT University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our error estimation. We introduce the expanding neural network method as a novel iterative regularization scheme and prove its regularization properties under different a priori assumptions about the exact solutions. Within this framework, the number of neurons serves as both the regularization parameter and iteration number. We demonstrate that for data with high noise levels, a small network architecture is sufficient to obtain a stable solution, whereas a larger architecture may compromise stability due to overfitting. Furthermore, under standard assumptions in regularization theory, we derive convergence rate results for neural networks in the context of variational regularization. Several numerical examples are presented to illustrate the robustness of the proposed neural network-based algorithms.

STDDM for the Helmholtz equation in inhomogeneous medium

Weiying Zheng Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Yuhao Wang

Abstract: In this talk, I will study the source-transfer-domain-decomposition method (STDDM) for solving high-frequency scattering problem of Helmholtz equation in a background medium with compact inhomogeneity. We first truncate the unbounded domain with the uniaxial perfectly matched layer (UPML) method. An wavenumber-explicit infsup condition is proved for the bilinear form of the truncated PML problem, where the infsup constant is shown to be $O(k\ln k)$, with $k$ being the wavenumber. Based on this infsup condition, we obtain preasymptotic finite element error estimates for the finite element approximation to the truncated problem. We propose an STDDM for solving the truncated PML problem and have proved the convergence of the method.