Inverse Problems

This work studies phase retrieval for wave fields, aiming to recover the phase of an incoming wave from multi-plane intensity measurements behind different types of linear and nonlinear media. We show that unique phase retrieval can be achieved by utilizing intensity data produced by multiple media. This uniqueness does not require prescribed boundary conditions for the phase in the incidence plane, in contrast to existing phase retrieval methods based on the transport of intensity equation. Moreover, the uniqueness proofs lead to explicit phase reconstruction algorithms. Numerical simulations are presented to validate the theory.

In this paper, we present the first uniqueness result on the bounded time inverse scattering problem for a semilinear Dirac equation with smooth nonlinearity F(x,z) where (x,z)∈R3×C4 and x is the spatial variable. We show that the solution map, which sends initial data at time 0 to the solution at time T, uniquely determines F(x,z) on x∈R3 and |z|⩽M, where M is a constant depend on the solution map, under the assumption that ∂zF(x,0) and ∂z2F(x,0) are known. In the proof, we construct a sequence of collisions approaching the initial timeline to simulate a boundary collision. This technique enables us to overcome the difficulties of this hyperbolic system without assumptions on the nonlinearity structure.

Traction force microscopy (TFM) is a method widely used in biophysics and cell biology to determine forces that biological cells apply to their environment. In the experiment, the cells adhere to a soft elastic substrate, which is then deformed in response to cellular traction forces. The inverse problem consists in computing the traction stress applied by the cell from microscopy measurements of the substrate deformations. In this work, we consider a linear model, in which 3D forces are applied at a 2D interface, called 2.5D TFM, and a nonlinear pure 2D model, from which we directly obtain a linear pure 2D model. All models lead to a linear resp. nonlinear parameter identification problem for a boundary value problem of elasticity. We analyze the respective forward operators and conclude with some numerical experiments for simulated and experimental data.

Iterative inversion of seismic, ultrasonic, and other wave data by local gradient-based optimization of mean-square data prediction error (full waveform inversion or FWI) can fail to converge to useful model estimates if started from an initial model predicting wave arrival times in error by more than half a wavelength (a phenomenon known as cycle skipping). Matched source waveform inversion (MSWI) extends the wave propagation model by a filter that shifts predicted waves to fit observed data. The MSWI objective adds a penalty for deviation of this filter from the identity to the mean-square data misfit. The extension allows the inversion to make large model adjustments while maintaining data fit and so reduces the chances of local optimization iterates stagnating at non-informative model estimates. Theory suggests that MSWI applied to acoustic transmission data with single-arrival wavefronts may produce an estimate of refractive index similar to the result of travel time inversion, but without requiring explicit identification of travel times. Numerical experiments conform to this expectation, in that MSWI applied to single arrival transmission data gives reasonable model estimates in cases where FWI fails. This MSWI model can then be used to jumpstart FWI for further refinement of the model. The addition of moderate amounts of noise (30%) does not negatively impact MSWI’s ability to converge. However, MSWI applied to data with multiple arrivals is no longer theoretically equivalent to travel-time tomography and exhibits the same tendency to cycle-skip as does FWI.

In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the convergence and approximation error of the proposed regularized inverse Born series is provided. The results show that the proposed series converges when the inverse Born approximations of the perturbations are sufficiently small. The preliminary numerical results show the capability of the proposed regularized inverse Born approximation and series for recovering the isotropic inhomogeneous media.

In two dimensions, we consider the problem of reconstructing a vector field from partial knowledge of its zeroth and first moment ray transforms. Different from existing works the data is known on a subset of lines, namely the ones intersecting a given arc. The problem is non-local and, for partial data, severely ill-posed. We present a reconstruction method which recovers the vector field in the convex hull of the arc. An algorithm based on this method is implemented on some numerical experiments. While still ill-posed the discretization stabilizes the numerical reconstruction.

Consider an inverse scattering of electromagnetic plane waves by an inhomogeneous penetrable conductive medium containing some unknown impenetrable obstacles. We establish the global uniqueness results for our inverse problem in determining the conductive medium D, the electric permittivity ε, the conductivity σ, the surface conductivity η, the internal embeddings D0 and its physical properties from the wave-field measurements corresponding to the incident electromagnetic plane waves. The global uniqueness proof is based on a new version on the a prior estimates for the magnetic wave-fields, the constructions of well-posed conductive transmission systems related to Maxwell equations in a suitable domain and various auxiliary Dirichlet boundary value problems in related annular domains. It is worthily mentioned that the simultaneous identification of (D, ε, σ, η, D0, B) for our inverse problem can be shown without knowing the a prior physical properties for the conductive medium D and its embeded objects D0.

We consider the Schrödinger equation with a multipoint potential of the Bethe−Peierls−Thomas−Fermi type. We show that such a potential in dimension d = 2 or d = 3 is uniquely determined by its scattering amplitude at a fixed positive energy. Moreover, we show that there is no non-zero potential of this type with zero scattering amplitude at a fixed positive energy and a fixed incident direction. Nevertheless, we also show that a multipoint potential of this type is not uniquely determined by its scattering amplitude at a positive energy E and a fixed incident direction. Our proofs also contribute to the theory of inverse source problem for the Helmholtz equation with multipoint source.

Computed Tomography(CT) is a highly effective inspection methodology for non destructive testing, however, it faces challenges with objects that have a significant aspect ratio. In such scenarios, computed laminography(CL) serves as a viable alternative. The unique geometric configuration of cone-beam rotary CL makes the classical FDK reconstruction algorithm unsuitable. To address this inverse problem, we proposed an approximate analytical reconstruction method inspired by the core idea of the FDK algorithm. Firstly, we addressed the cone-beam rotary CL reconstruction problem by approximating it as a series of fan-beam reconstruction tasks, and deriving the geometric relationships between the original coordinate system and the fan-beam coordinates. Secondly, the fan-beam reconstruction formula within the fan-beam coordinate system is derived. Finally, disregarding the fact that the projection data come from different planes based on the idea of the FDK algorithm, an approximate analytical reconstruction algorithm is derived based on the geometric relationships between the original coordinate system and the detector coordinates. The effectiveness of our method is demonstrated through simulations and real data experiments. The results demonstrate that the proposed method mitigates the superimposition artifacts and enhances the fine structural details to a certain extent compared to the CL-FBP method, while preserving the object boundaries.

A version of the globally convergent convexification numerical method is constructed for the problem of electrical impedance tomography in the 2D case. An important element of this version is the presence of the viscosity term. Global convergence analysis is carried out. Results of numerical experiments are presented.

This paper proposes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier–Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works.

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in ”quasi real-time”. We illustrate these results with numerical experiments.

The identification of time-dependent parameters in partial differential equations is a fundamental challenge in inverse problems, with wide-ranging implications in fields such as quantum mechanics, wave propagation, and material science. In this paper, we investigate the inverse problem of stably recovering a time-dependent matrix-valued potential in the context of the wave equation. We consider a cylindrical domain with measurements taken on an arbitrary subset of the lateral surface. The matrix potential plays a crucial role in the regulation of wave dynamics and energy propagation, making its accurate reconstruction essential to understand complex physical phenomena. We establish a logarithmic stability estimate for determining this matrix potential from the partial Dirichlet-to-Neumann map. Our results contribute to the theoretical understanding of inverse problems and provide valuable insight into applications such as inverse scattering and time-dependent quantum field theory.

Recently, the stochastic asymptotical regularization (SAR) has been developed in Zhang and Chen (2023 Inverse Problems 39 015007) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this paper, we extend the regularization theory of SAR for nonlinear inverse problems. By combining techniques from classical regularization theory and stochastic analysis, we prove the regularizing properties of SAR with regard to mean-square convergence. The convergence rate results under the canonical sourcewise condition are also studied. Several numerical examples are used to show the accuracy and advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can quantify the uncertainty in error estimates for ill-posed problems, improve accuracy by selecting the optimal path, escape local minima for nonlinear problems, and identify multiple solutions by clustering samples of obtained approximate solutions.

Recovering a low-complexity signal from its noisy observations by regularization methods is a cornerstone of inverse problems and compressed sensing. Stable recovery ensures that the original signal can be approximated linearly by optimal solutions of the corresponding Morozov or Tikhonov regularized optimization problems. In this paper, we propose new characterizations for stable recovery in finite-dimensional spaces, uncovering the role of nonsmooth second-order information. These insights enable a deeper understanding of stable recovery and their practical implications. As a consequence, we apply our theory to derive new sufficient conditions for stable recovery of the analysis group sparsity problems, including the group sparsity and isotropic total variation problems. Numerical experiments on these two problems give favorable results about using our conditions to test stable recovery.

In this paper, we study the heat equation on a tree-shaped network with a piecewise regular diffusion coefficient. By developing new Carleman estimates, we establish stability results for the identification of the diffusion coefficient. These stability estimates are derived using either internal measurements or boundary observations, offering robust insights into the inverse problem for this class of equations.

In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequentially initializing the lower-level problem, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Moreover, by collecting the lower-level trajectory, we uncover an interesting connection to the incremental load method. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. We moreover prove that the proposed tangential cone condition is satisfied.

In image reconstruction, an accurate quantification of uncertainty is of great importance for informed decision making. Here, the Bayesian approach to inverse problems can be used: the image is represented through a random function that incorporates prior information which is then updated through Bayes’ formula. However, finding a prior is difficult, as images often exhibit non-stationary effects and multi-scale behavior. Thus, usual Gaussian process priors are not suitable. Deep Gaussian processes, on the other hand, encode non-stationary behavior in a natural way through their hierarchical structure. To apply Bayes’ formula, one commonly employs a Markov chain Monte Carlo (MCMC) method. In the case of deep Gaussian processes, sampling is especially challenging in high dimensions: the associated covariance matrices are large, dense, and changing from sample to sample. A popular strategy towards decreasing computational complexity is to view Gaussian processes as the solutions to a fractional stochastic partial differential equation (SPDE). In this work, we investigate efficient computational strategies to solve the fractional SPDEs occurring in deep Gaussian process sampling, as well as MCMC algorithms to sample from the posterior. Namely, we combine rational approximation and a determinant-free sampling approach to achieve sampling via the fractional SPDE. We test our techniques in standard Bayesian image reconstruction problems: upsampling, edge detection, and computed tomography. In these examples, we show that choosing a non-stationary prior such as the deep GP over a stationary GP can improve the reconstruction. Moreover, our approach enables us to compare results for a range of fractional and non-fractional regularity parameter values.

The sensor placement problem for stochastic linear inverse problems consists of determining the optimal manner in which sensors can be employed to collect data. Specifically, one wishes to place a limited number of sensors over a large number of candidate locations, quantifying and optimizing over the effect this data collection strategy has on the solution of the inverse problem. In this article, we provide a global optimality condition for the sensor placement problem via a subgradient argument, obtaining sufficient and necessary conditions for optimality, and marking certain sensors as dominant or redundant, i.e. always on or always off. We demonstrate how to take advantage of this optimality criterion to find approximately optimal binary designs, i.e. designs where no fractions of sensors are placed. Leveraging our optimality criteria, we derive a powerful low-rank formulation of the A-optimal design objective for finite element-discretized function space settings, demonstrating its high computational efficiency, particularly in terms of derivatives, and study globally optimal designs for a Helmholtz-type source problem and extensions towards optimal binary designs.

Electroencephalograms (EEG) are invaluable for treating neurological disorders, however, mapping EEG electrode readings to brain activity requires solving a challenging inverse problem. For time series data, the use of ℓ1 regularization quickly becomes intractable for many solvers, and, despite the reconstruction advantages of ℓ1 regularization, ℓ2-based approaches such as standardized low-resolution brain electromagnetic tomography sLORETA are used in practice. In this work, we formulate EEG source localization as a graphical generalized elastic net inverse problem and present a variable projected augmented Lagrangian algorithm (VPAL) suitable for fast EEG source localization. We prove convergence of this solver for a broad class of separable convex, potentially non-smooth functions subject to linear constraints. Leveraging the efficiency of the proposed VPAL algorithm, we introduce a windowed variation, VPAL W , that computes time dynamics in sequence suitable for real-time reconstruction. Our proposed methods are compared to state-of-the-art approaches including sLORETA and other methods for ℓ1-regularized inverse problems.

We focus on Bayesian inverse problems with Gaussian likelihood, linear forward model, and priors that can be formulated as a Gaussian mixture. Such a mixture is expressed as an integral of Gaussian density functions weighted by a mixing density over the mixing variables. Within this framework, the corresponding posterior distribution also takes the form of a Gaussian mixture, and we derive the closed-form expression for its posterior mixing density. To sample from the Gaussian posterior mixture, we propose to use the marginal then conditional sampler, which comprises two steps: First, we sample the mixing variables from the posterior mixing density, then we sample the variables of interest from Gaussian densities conditioned on the sampled mixing variables. However, the posterior mixing density is relatively difficult to sample from, especially in high dimensions. Therefore, we propose to replace the posterior mixing density by a dimension-reduced approximation, and we provide a bound in the Hellinger distance for the resulting approximate posterior. We apply the proposed approach to a posterior with Laplace prior, where we introduce two dimension-reduced approximations for the posterior mixing density. Our numerical experiments indicate that samples generated via the proposed approximations have very low correlation and are close to the exact posterior.

This paper concerns the inverse random potential scattering problem for the Helmholtz equation in one dimension. The random potential is assumed to be a generalized microlocally isotropic Gaussian random field, whose covariance is a classical pseudodifferential operator. To investigate the inverse problem, as an effective mathematical tool, the scattering theory is employed to obtain an analytic domain and estimates for the resolvent of the elliptic operator with rough potentials. For the inverse problem, based on the resolvent estimates, we show that the principal symbol of the covariance operator can be reconstructed by a single realization of the boundary data averaged over the high-frequency band almost surely. Consequently, by analyticity of the resolvent, the uniqueness of the inverse problem can be achieved by data at multiple frequencies only in a finite interval. The method developed here is unified and can be applied to other stochastic inverse scattering problems in higher dimensions for various wave equations by boundary measurements.

We consider the efficient numerical minimization of Tikhonov functionals with nonlinear operators and non-smooth and non-convex penalty terms, which appear for example in variational regularization. For this, we consider a new class of SCD semismooth* Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence. We present a detailed description of these methods, and provide explicit algorithms in the case of sparsity and total-variation penalty terms. The numerical performance of these methods is then illustrated on a number of tomographic imaging problems.

Linear inverse problems (LIPs) have been a pivotal concern in the fields of signal and image processing, where sparse linear inverse problems (SLIPs) emerge as a prominent research topic. The aim of SLIPs is to recover a K-sparse vector \x from an underdetermined linear system. While existing sparse recovery methods are effective, more efficient and effective algorithms are still needed. To this end, we develop the Heavy-ball Enhanced Pseudo-Inverse-Based Hard Thresholding (HBPHT) algorithm. By introducing the idea of subspace pursuit to HBPHT, we further propose the Heavy-ball Enhanced Pseudo-Inverse-Based Hard Thresholding Pursuit (HBPHP) algorithm. To ensure the effectiveness of the proposed methods, we prove that HBPHT and HBPHP exhibit linear convergence to the true solution under certain sufficient conditions based on restricted isometry property (RIP). Extensive tests based on synthetic and real-world data demonstrate the superiority of our approaches over the state-of-the-art sparse recovery methods. (1) In simulated experiments, when sparsity level K is relatively large, HBPHP demonstrates a success recovery probability that is two to four times higher than that of heavy-ball-based hard thresholding pursuit (HBHTP), which has superb recovery performance. (2) In image reconstruction tasks, HBPHP consistently achieves PSNR values that are at least 0.3 to 1 dB higher than those of HBHTP. (3) For face recognition applications, HBPHP outperforms HBHTP with an average improvement of 3 percentage points in recognition rate.

This paper addresses a general multi-dimensional time-fractional sideways problem, with a focus on determining spatial derivatives of the temperature distribution from internally measured data. The problem is known to be severely ill-posed, as its solution does not depend continuously on the data. By selecting a suitable spectral source set, we establish lower bounds on the worst-case error associated with the problem. Furthermore, we demonstrate that a general Tikhonov regularization method can achieve the optimal convergence rates derived from our analysis. To the best of our knowledge, this represents a new optimal result for the multi-dimensional sideways problem. Numerical experiments are included to validate the theoretical results.