Table of contents

In this work, we consider the inverse scattering problem of determining an unknown refractive index from the far-field measurements using the nonparametric Bayesian approach. We use a collection of large 'samples', which are noisy discrete measurements taking from the scattering amplitude. We will study the frequentist property of the posterior distribution as the sample size tends to infinity. Our aim is to establish the consistency of the posterior distribution with an explicit contraction rate in terms of the sample size. We will consider two different priors on the space of parameters. The proof relies on the stability estimates of the forward and inverse problems. Due to the ill-posedness of the inverse scattering problem, the contraction rate is of a logarithmic type. We also show that such contraction rate is optimal in the statistical minimax sense.

This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.

This paper follows a detection-theoretic approach for using synthetic-aperture measurements, made at multiple moving passive receivers, in order to form an image showing the locations of stationary sources that are radiating unknown electromagnetic or acoustic waves. The paper starts with a physics-based model for the propagating fields, and, following the general approach of McWhorter et al (2023 arXiv:2302.06816, IEEE Open J. Signal Process.4 437–51), derives a detection statistic that is used for the image formation. This detection statistic is a quadratic function of the data. Each point in the scene is tested as a possible hypothesized location for a source, and the detection statistic is plotted as a function of location. Because this image formation process is nonlinear, the standard linear methods for determining resolution cannot be applied. This paper shows how to analyze the detection image by first writing the noiseless image as a coherent sum of shifted complex ambiguity functions of the source waveform. The paper then develops a technique for calculating image resolution; resolution is found to depend on the sensor-source geometry and also on the properties (bandwidth and temporal duration) of the source waveform. Optimal filtering of the image is given, but a simple example suggests that optimal filtering may have little effect. Analysis is also given for the case in which multiple sources are present.

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle's geometric configurations and the wave field's vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.

Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schrödinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions (DFDs) has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered DFDs with general non-linear filters. In particular, our results generalize singular value decomposition-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

In this article, we study the problem of recovering sparse spikes with over-parametrized projected descent. We first provide a theoretical study of approximate recovery with our chosen initialization method: Continuous Orthogonal Matching Pursuit without Sliding. Then we study the effect of over-parametrization on the gradient descent which highlights the benefits of the projection step. Finally, we show the improved calculation times of our algorithm compared to state-of-the-art model-based methods on realistic simulated microscopy data.

This work develops a machine learned structural design model for continuous beam systems from the inverse problem perspective. After demarcating between forward, optimisation and inverse machine learned operators, the investigation proposes a novel methodology based on the recently developed influence zone concept which represents a fundamental shift in approach compared to traditional structural design methods. The aim of this approach is to conceptualise a non-iterative structural design model that predicts cross-section requirements for continuous beam systems of arbitrary system size. After generating a dataset of known solutions, an appropriate neural network architecture is identified, trained, and tested against unseen data. The results show a mean absolute percentage testing error of 1.6% for cross-section property predictions, along with a good ability of the neural network to generalise well to structural systems of variable size. The CBeamXP dataset generated in this work and an associated python-based neural network training script are available at an open-source data repository to allow for the reproducibility of results and to encourage further investigations.

In this paper we study a wave equation with discontinuous principal coefficient within a bounded domain of arbitrary dimension. It is obtained the stability of the inverse problem of recovering a space-dependent coefficient by observing a trace of the corresponding solution on part of the boundary. We provide a precise estimate of the minimum required time, as a function of the velocity change and domain size. The main tools are new global Carleman estimates for the transmission system with a particular weight function adapted to the interface geometry, which allows to obtain an optimal estimate of the minimum time.

Microtexture regions (MTRs) are collections of grains with similar crystallographic orientation. Because their presence in titanium alloys can significantly impact aerospace component life, a nondestructive method to detect and characterize MTR is needed. In this work, we propose to use data from two nondestructive evaluation methods, eddy current testing (ECT) and scanning acoustic microscopy (SAM), in order to recover the boundary and dominant crystallographic orientation of each MTR in a specimen. ECT is an electromagnetic method that is sensitive to changes in crystallographic orientation associated with MTR; however, its low resolution prevents it from resolving MTR boundaries well. In contrast, SAM is a high frequency ultrasound method that is able to resolve MTR boundaries but is not sensitive to orientation. This paper proposes an algorithm to characterize MTR that makes use of a method known as covariance generalized matching component analysis. This method is used to build a surrogate linear forward model that relates MTR boundaries and orientation to ECT data. The model is inverted using the SAM data as a structural prior. We demonstrate this technique using simulated ECT and experimental SAM data from a large grain titanium specimen.

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Magnetic resonance imaging (MRI) is a widely used medical imaging technique, but its long acquisition time can be a limiting factor in clinical settings. To address this issue, researchers have been exploring ways to reduce the acquisition time while maintaining the reconstruction quality. Previous works have focused on finding either sparse samplers with a fixed reconstructor or finding reconstructors with a fixed sampler. However, these approaches do not fully utilize the potential of joint learning of samplers and reconstructors. In this paper, we propose an alternating training framework for jointly learning a good pair of samplers and reconstructors via deep reinforcement learning. In particular, we consider the process of MRI sampling as a sampling trajectory controlled by a sampler, and introduce a novel sparse-reward partially observed Markov decision process (POMDP) to formulate the MRI sampling trajectory. Compared to the dense-reward POMDP used in existing works, the proposed sparse-reward POMDP is more computationally efficient and has a provable advantage. Moreover, the proposed framework, called learning to sample and reconstruct (L2SR), overcomes the training mismatch problem that arises in previous methods that use dense-reward POMDP. By alternately updating samplers and reconstructors, L2SR learns a pair of samplers and reconstructors that achieve state-of-the-art reconstruction performances on the fastMRI dataset. Codes are available at https://github.com/yangpuPKU/L2SR-Learning-to-Sample-and-Reconstruct.

The retrieval of non-Born scatterers is addressed within the contrast source inversion (CSI) framework by means of a novel multi-step inverse scattering method that jointly exploits prior information on the class of targets under investigation and progressively-acquired knowledge on the domain under investigation. The multi-resolution (MR) representation of the unknown contrast sources is iteratively retrieved by applying a Bayesian compressive sensing (BCS) sparsity-promoting approach based on a constrained relevance vector machine solver. Representative examples of inversions from synthetic and experimental data are reported to give some indications on the reliability and the robustness of the proposed MR-BCS-CSI method. Comparisons with recent and competitive state-of-the-art alternatives are reported, as well.