arXiv subjects

Lingyun Qiu

Publications and source records attributed to Lingyun Qiu.

At least 19 recordsLinked to original sources

arXiv2026

Sediment Concentration Estimation via Multiscale Inverse Problem and Stochastic Homogenization

We develop a multiscale framework for estimating sediment concentration in water flow from acoustic wave measurements. At the microscopic scale, the sediment distribution is modeled by a spatially inhomogeneous Poisson cloud, while the quantity of interest is its macroscopic concentration. For the associated random wave model, we derive an effective medium whose coefficient is explicitly related to the local probability of sediment occurrence. This effective description avoids resolving individual sediment particles and provides a computationally tractable forward model for inversion. We then formulate the recovery of the effective medium, and hence the sediment concentration, as an inverse medium problem from partial boundary measurements, and investigate numerical strategies including model mollification and shot averaging. Numerical experiments demonstrate that the effective model captures the macroscopic wave behavior and can be used to obtain accurate estimates of sediment concentration.

math.NA↗

arXiv2026

Magneto-Acousto-Electric Tomography with Magnetic Field Measurements: Modeling, Inversion and Stability

Magneto-acousto-electric tomography (MAET) combines ultrasound with a static magnetic field to infer the electrical conductivity of an object. In this paper, we present a rigorous quasi-static mathematical model for MAET with magnetic field measurements and introduce an adjoint problem to decouple the resulting hybrid inverse problem. This yields a two-step inversion procedure: solving an acoustic inverse source problem and then recovering the conductivity from an internal current density. For the second step, by exploiting the analytic structure of a coil-determined field, we establish an interior Hölder stability estimate without imposing a pointwise nonzero constraint on the internal data. We further prove that, under explicit smallness assumptions on the conductivity and coil geometry, the conductivity can be recovered with region-of-interest Lipschitz stability in both bounded and half-space geometries.

math.AP↗

arXiv2026

Non-line-of-sight imaging with arbitrary relay surface geometries via 3D Gaussian Transient Rendering

Imaging objects hidden outside the direct line of sight expands the effective field of view and is critical for applications such as autonomous driving and robotic perception. Despite impressive progress in time-of-flight (ToF)-based non-line-of-sight (NLOS) imaging, real-world deployment remains challenging because practical measurements are often collected over spatially limited, arbitrarily shaped relay regions-conditions that violate the planar-wall and dense-sampling assumptions made by most existing methods. To address these limitations, we propose a LOS-guided NLOS imaging pipeline that imposes no geometric assumptions on the relay surface and naturally supports both confocal and non-confocal configurations. Our method represents the hidden scene using 3D Gaussian primitives and couples them with an efficient, differentiable transient rendering model, enabling end-to-end optimization directly from measured transients. We validate our approach on real-world measurements from both a public dataset and a custom-built capture system. Across settings, our method achieves state-of-the-art reconstruction fidelity under spatially limited, sparsely sampled conditions, and significantly outperforms existing methods on complex, arbitrary relay surface geometries.

physics.optics↗

arXiv2025

Optimal-Transport Stability of Inverse Point-Source Problems for Elliptic and Parabolic Equations

We establish quantitative global stability estimates, formulated in terms of optimal transport (OT) cost, for inverse point-source problems governed by elliptic and parabolic equations with spatially varying coefficients. The key idea is that the Kantorovich dual potential can be represented as a boundary functional of suitable adjoint solutions, thereby linking OT geometry with boundary observations. In the elliptic case, we construct complex geometric optics solutions that enforce prescribed pointwise constraints, whereas in the parabolic case we employ controllable adjoint solutions that transfer interior information to the boundary. Under mild regularity and separation assumptions, we obtain estimates of the form \[ \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ)} \quad \text{and} \quad \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ\times[0,T])}, \] where $μ$ and $ν$ are admissible point-source measures. These results provide a unified analytical framework connecting inverse source problems and optimal transport, and establish OT-based stability theory for inverse source problems governed by partial differential equations with spatially varying coefficients.

math.NA↗

arXiv2025

Fast and Memory-efficient Non-line-of-sight Imaging with Quasi-Fresnel Transform

Non-line-of-sight (NLOS) imaging seeks to reconstruct hidden objects by analyzing reflections from intermediary surfaces. Existing methods typically model both the measurement data and the hidden scene in three dimensions, overlooking the inherently two-dimensional nature of most hidden objects. This oversight leads to high computational costs and substantial memory consumption, limiting practical applications and making real-time, high-resolution NLOS imaging on lightweight devices challenging. In this paper, we introduce a novel approach that represents the hidden scene using two-dimensional functions and employs a Quasi-Fresnel transform to establish a direct inversion formula between the measurement data and the hidden scene. This transformation leverages the two-dimensional characteristics of the problem to significantly reduce computational complexity and memory requirements. Our algorithm efficiently performs fast transformations between these two-dimensional aggregated data, enabling rapid reconstruction of hidden objects with minimal memory usage. Compared to existing methods, our approach reduces runtime and memory demands by several orders of magnitude while maintaining imaging quality. The substantial reduction in memory usage not only enhances computational efficiency but also enables NLOS imaging on lightweight devices such as mobile and embedded systems. We anticipate that this method will facilitate real-time, high-resolution NLOS imaging and broaden its applicability across a wider range of platforms.

cs.CV↗

arXiv2025

Uniqueness of the inverse source problem for fractional diffusion-wave equations

This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data is collected subsequent to an incident. We establish the uniqueness of either the spatial or the temporal component of the source, provided that the temporal component exhibits an asymptotic expansion at infinity. Taking anomalous diffusion as a typical example, we gather the asymptotic behavior of one of the following quantities: the concentration on partial interior region or at a point inside the region, or the flux on partial boundary or at a point on the boundary. The proof is based on the asymptotic expansion of the solution to the fractional diffusion-wave equation. Notably, our approach does not rely on the conventional vanishing conditions for the source components. We also observe that the extent of uniqueness is dependent on the fractional order.

math.NA↗

arXiv2025

An Acoustic Inversion-Based Flow Measurement Model in 3D Hydrodynamic Systems

This study extends the flow measurement method initially proposed in [22] to three-dimensional scenarios, addressing the growing need for accurate and efficient non-contact flow measurement techniques in complex hydrodynamic environments. Compared to conventional Acoustic Doppler Current Profilers (ADCPs) and remote sensing-based flow monitoring, the proposed method enables high-resolution, continuous water velocity measurement, making it well-suited for hazardous environments such as floods, strong currents, and sediment-laden rivers. Building upon the original approach, we develop an enhanced model that incorporates multiple emission directions and flexible configurations of receivers. These advancements improve the adaptability and accuracy of the method when applied to three-dimensional flow fields. To evaluate its feasibility, we conducted extensive numerical simulations designed to mimic real-world hydrodynamic conditions. The results demonstrate that the proposed method effectively handles diverse and complex flow field configurations, highlighting its potential for practical applications in water resource management and hydraulic engineering.

math.NA↗

arXiv2024

Neumann Series-based Neural Operator for Solving Inverse Medium Problem

The inverse medium problem, inherently ill-posed and nonlinear, presents significant computational challenges. This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance, even with varying scattering properties and noisy data. The robustness and adaptability of our framework provide crucial insights and methodologies, extending its applicability to a broad spectrum of scattering problems. These advancements mark a significant step forward in the field, offering a scalable solution to traditionally complex inverse problems.

math-ph↗

arXiv2023

Traceability of Water Pollution: An Inversion Scheme Via Dynamic Complex Geometrical Optics Solutions

We investigate the identification of the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Employing the concept of dynamic complex geometrical optics (CGO) solutions, a variational formulation of the inverse source problem is analyzed, leading to a proof of uniqueness result. Our proposed two-step reconstruction algorithm first determines the point source locations and subsequently reconstructs the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that the proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, by partitioning the algorithm into online and offline computations, and concentrating computational demand offline, real-time pollutant traceability becomes feasible. This method, applicable in various fields - especially those related to water pollution, can identify the source of a contaminant in the environment, thus serving as a valuable tool in environmental protection.

math.NA↗

arXiv2022

Flow Measurement: An Inverse Problem Formulation

This paper proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on a few boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. Our formulation is capable of simulating a variety of practical measurement scenarios.

math.OC↗

arXiv2022

Few-shot Non-line-of-sight Imaging with Signal-surface Collaborative Regularization

The non-line-of-sight imaging technique aims to reconstruct targets from multiply reflected light. For most existing methods, dense points on the relay surface are raster scanned to obtain high-quality reconstructions, which requires a long acquisition time. In this work, we propose a signal-surface collaborative regularization (SSCR) framework that provides noise-robust reconstructions with a minimal number of measurements. Using Bayesian inference, we design joint regularizations of the estimated signal, the 3D voxel-based representation of the objects, and the 2D surface-based description of the targets. To our best knowledge, this is the first work that combines regularizations in mixed dimensions for hidden targets. Experiments on synthetic and experimental datasets illustrated the efficiency and robustness of the proposed method under both confocal and non-confocal settings. We report the reconstruction of the hidden targets with complex geometric structures with only $5 \times 5$ confocal measurements from public datasets, indicating an acceleration of the conventional measurement process by a factor of 10000. Besides, the proposed method enjoys low time and memory complexities with sparse measurements. Our approach has great potential in real-time non-line-of-sight imaging applications such as rescue operations and autonomous driving.

cs.CV↗

arXiv2022

Non-line-of-sight imaging with arbitrary illumination and detection pattern

Non-line-of-sight (NLOS) imaging aims at reconstructing targets obscured from the direct line of sight. Existing NLOS imaging algorithms require dense measurements at rectangular grid points in a large area of the relay surface, which severely hinders their availability to variable relay scenarios in practical applications such as robotic vision, autonomous driving, rescue operations and remote sensing. In this work, we propose a Bayesian framework for NLOS imaging with no specific requirements on the spatial pattern of illumination and detection points. By introducing virtual confocal signals, we design a confocal complemented signal-object collaborative regularization (CC-SOCR) algorithm for high quality reconstructions. Our approach is capable of reconstructing both albedo and surface normal of the hidden objects with fine details under the most general relay setting. Moreover, with a regular relay surface, coarse rather than dense measurements are enough for our approach such that the acquisition time can be reduced significantly. As demonstrated in multiple experiments, the new framework substantially enhances the applicability of NLOS imaging.

eess.IV↗

arXiv2020

Analysis of Seismic Inversion with Optimal Transportation and Softplus Encoding

This paper is devoted to theoretical and numerical investigation of the local minimum issue in seismic full waveform inversion (FWI). This paper provides a mathematical analysis of optimal transportation (OT) type objective function's differentiability and proves that the gradient obtained in the adjoint-state method does not depend on the particular choice of the Kantorovich potentials. A novel approach using the softplus encoding method is presented to generalize and impose the OT metric on FWI. This approach improves the convexity of the objective function and mitigates the cycle-skipping problem. The effectiveness of the proposed method is demonstrated numerically on an inversion task with the benchmark Marmousi model.

math.NA↗

arXiv2017

Determining anisotropic conductivity using diffusion tensor imaging data in magneto-acoustic tomography with magnetic induction

In this paper we present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor by integrating magneto-acoutic tomography with data acquired from diffusion tensor imaging. Magneto-acoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid, non-invasive medical imaging technique to produce conductivity images with improved spatial resolution and accuracy. Diffusion Tensor Imaging (DTI) is also a non- invasive technique for characterizing the diffusion properties of water molecules in tissues. We propose a model for anisotropic conductivity in which the conductivity is proportional to the diffusion tensor. Under this assumption, we propose an optimal control approach for reconstructing the anisotropic electrical conductivity tensor. We prove convergence and Lipschitz type stability of the algorithm and present numerical examples to illustrate its accuracy and feasibility.

math.NA↗

arXiv2016

Applications of CGO Solutions on Coupled-Physics Inverse Problems

This paper surveys inverse problems arising in several coupled-physics imaging modalities for both medical and geophysical purposes. These include Photo-acoustic Tomography (PAT), Thermo-acoustic Tomography (TAT), Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric Tomography (AET). These inverse problems typically consists of multiple inverse steps, each of which corresponds to one of the wave propagations involved. The review focus on those steps known as the inverse problems with internal data, in which the complex geometrical optics (CGO) solutions to the underlying equations turn out to be useful in showing the uniqueness and stability in determining the desired information.

math.AP↗

arXiv2015

Analysis of the Magneto-acoustic Tomography with Magnetic Induction (MAT-MI)

Magnetoacoustic tomography with magnetic induction (MAT-MI) is a coupled-physics medical imaging modality for determining conductivity distribution in biological tissue. The capability of MAT-MI to provide high resolution images has been demonstrated experimentally. MAT-MI involves two steps. The first step is a well-posed inverse source problem for acoustic wave equation, which has been well studied in the literature. This paper concerns mathematical analysis of the second step, a quantitative reconstruction of the conductivity from knowledge of the internal data recovered in the first step, using techniques such as time reversal. The problem is modeled by a system derived from Maxwell's equations. We show that a single internal data determines the conductivity. A global Lipschitz type stability estimate is obtained. A numerical approach for recovering the conductivity is proposed and results from computational experiments are presented.

math.NA↗

arXiv2014

Generalized Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting. We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer, and generalize existing convergence rates results. The key tool in our analysis is a novel source condition, which we put into relation to the existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces.

math.FA↗

arXiv2014

Inverse boundary value problem for the Helmholtz equation: Multi-level approach and iterative reconstruction

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing discontinuities. A conditional Lipschitz stability estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce hierarchical compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an optimal bound of the stability constant, which leads to a condition on the compression rate pertaining to these approximations.

math.NA↗