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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.

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., Comment: 21 pages

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.

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.

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., Comment: main article: 10 pages, 7 figures supplement: 11 pages, 24 figures

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., Comment: main article: 32 pages with 8 figures; supplementary information: 49 pages with 26 figures

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.

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.

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., Comment: 29 pages

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.

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., Comment: 16 pages

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.

We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a closed, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of closed, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps.

In this paper we study the inverse boundary value problem of determining the potential in the Schr\"{o}dinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems.