Nonlinear Ill-posed Problems Research Articles

Microwave Near-Field Dynamical Tomography of Thorax at Pulmonary and Cardiovascular Activity.

The developed near-field microwave diagnostics of dynamical lung tomography provide information about variations of air and blood content depth structure in the processes of breathing and heart beating that are unattainable for other available methods. The method of dynamical pulse 1D tomography (profiling) is based on solving the corresponding nonlinear ill-posed inverse problem in the extremely complicated case of the strongly absorbing frequency-dispersive layered medium with the dual regularization method-a new Lagrange approach in the theory of ill-posed problems. This method has been realized experimentally by data of bistatic measurements with two electrically small bow-tie antennas that provide a subwavelength resolution. The proposed methods of 3D lung tomography based on the multisensory pulse, multifrequency, or multi-base measurements are based on solving the corresponding integral equations in the Born approximation. The experimental 3D tomography of lung air content was obtained by the results of the multiple 1D pulse profiling by pulse measurements in several grid points over the planar square region of the thorax. Additionally, the possible applicability of multifrequency measurements of scattered harmonic signals in the monitoring of lungs was demonstrated by four-frequency measurements in the process of breathing. The results demonstrated the feasibility of the proposed control in the diagnosis of some lung diseases.

A Study of an Iteratively-Regularized Simplified Landweber Iteration for Nonlinear Inverse Problems in Hilbert Spaces

Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method.

Magnetic Induction Tomography: Separation of the Ill-Posed and Non-Linear Inverse Problem into a Series of Isolated and Less Demanding Subproblems.

Magnetic induction tomography (MIT) is based on remotely excited eddy currents inside a measurement object. The conductivity distribution shapes the eddies, and their secondary fields are detected and used to reconstruct the conductivities. While the forward problem from given conductivities to detected signals can be unambiguously simulated, the inverse problem from received signals back to searched conductivities is a non-linear ill-posed problem that compromises MIT and results in rather blurry imaging. An MIT inversion is commonly applied over the entire process (i.e., localized conductivities are directly determined from specific signal features), but this involves considerable computation. The present more theoretical work treats the inverse problem as a non-retroactive series of four individual subproblems, each one less difficult by itself. The decoupled tasks yield better insights and control and promote more efficient computation. The overall problem is divided into an ill-posed but linear problem for reconstructing eddy currents from given signals and a nonlinear but benign problem for reconstructing conductivities from given eddies. The separated approach is unsuitable for common and circular MIT designs, as it merely fits the data structure of a recently presented and planar 3D MIT realization for large biomedical phantoms. For this MIT scanner, in discretization, the number of unknown and independent eddy current elements reflects the number of ultimately searched conductivities. For clarity and better representation, representative 2D bodies are used here and measured at the depth of the 3D scanner. The overall difficulty is not substantially smaller or different than for 3D bodies. In summary, the linear problem from signals to eddies dominates the overall MIT performance.

Deep-plug-and-play proximal Gauss-Newton method with applications to nonlinear, ill-posed inverse problems

In this paper we propose a proximal Gauss-Newton method for the penalized nonlinear least squares optimization problem arising from regularization of ill-posed nonlinear inverse problems. By exploiting the modular structure that characterizes the proximal-type methods, we plug in a pre-trained graph neural net denoiser in place of the standard proximal map. This allows to mould the prior on the data. An encoder-decoder Graph U-Net architecture is proposed as denoiser, which works on unstructured data; its mathematical formulation is derived to analyse the Liptschitz condition. With the intent of showing the benefits of applying deep Plug-and-Play reconstructions, we consider as an exemplar application, the nonlinear Electrical Impedance Tomography, a promising non-invasive imaging technique mathematically formulated as a highly nonlinear ill-posed inverse problem.

New results for variational regularization with oversmoothing penalty term in Banach spaces

In this article on variational regularization for ill-posed nonlinear problems, we are once again discussing the consequences of an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales, but rarely in a Banach space setting. Our present results try to establish a theoretical justification of oversmoothing regularization in Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for Hölder-type smoothness and low order-type smoothness. An illustrative example is intended to indicate the specificity of occurring non-reflexive Banach spaces.

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess.These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

A generalized adaptive Levenberg–Marquardt method for solving nonlinear ill-posed problems

We investigate the method of regularization for the stable approximate solution to nonlinear ill-posed problems whose forward operators may not be Gâteaux differentiable. The method is designed by combining the classical Levenberg–Marquardt method with the two-point gradient iteration, and the adaptive stepsize which is related to the Tikhonov regularization parameter and the structure of the forward operator. In order to further enhance the acceleration effect, we employ a modified discrete backtracking search algorithm to determine the combination parameters involved. With the help of the concept of asymptotic stability and a generalized tangential cone condition, the convergence analysis of the proposed method is studied. Moreover, several numerical experiments are performed to illustrate the effectiveness and acceleration effect.

A modified steepest descent method for solving non-smooth inverse problems

The classical steepest descent method is applicable only on the inverse problems for which the forward operator is Gâteaux differentiable. In this paper, we propose an extension of the classical steepest descent method so that it can be used for solving non-smooth nonlinear ill-posed problems. Basically, we incorporate the Bouligand subderivative of the forward mapping in order to propose the extension. We study the convergence analysis of the proposed method by assuming the boundedness of the Bouligand subderivative as well as a modified tangential cone condition. In addition to this, we discuss an important feature, i.e., strongly convergent regularizing nature of the proposed method. Finally, we provide the numerical simulations to show the practicality of the proposed method and compare our results with that of existing methods in the literature.

SAR-CGAN: Improved generative adversarial network for EIT reconstruction of lung diseases

The image reconstruction of electrical impedance tomography (EIT) is a nonlinear ill-posed inverse problem, and the reconstructed images tend to have artifacts due to noise in the measurement system. The shallow neural networks are difficult to fully express the nonlinear relationship of the EIT imaging problems. Deep neural networks have demonstrated great potential to remove artifacts from initial conductivity images caused by system noise. In the paper, a novel deep learning method, namely, conditional generative adversarial network with soft-attention gates (AGs) and residual learning (SAR-CGAN), is proposed to solve the above problems of EIT imaging. SAR-CGAN consists of generator and discriminator. Specifically, the AGs are employed by the generator to improve sensitivity and accuracy for lung EIT imaging by suppressing feature activation in irrelevant EIT region. And residual learning framework is introduced in the discriminator to address the problem of model degradation and easily enjoy accuracy gains from greatly increased depth. Extensive confirmatory and comparative experiments with three kinds of lung disease are conducted and compared with CNN, DnCNN, Attention U-Net and CGAN methods. The imaging results indicate that SAR-CGAN method can accurately recover the position and boundaries of lesions in lung images and effective reconstruction can be accomplished for the new conductivity distribution and noisy samples.

Convergence results of nonlinear problems based on Tikhonov regularization method

In this paper, we study the convergent results of Tikhonov regularization method for solving nonlinear ill-posed problems with noisy operator in Banach spaces. And we give the convergent rates while both the right-hand side of the equation and the operator are given approximately errors. Furthermore we prove that the convergent rates depend on the interplay of the solution smoothness and the nonlinearity structure, and obtain concisely variational inequalities results about them.

Optimum trajectory learning in musculoskeletal systems with model predictive control and deep reinforcement learning

From the computational point of view, musculoskeletal control is the problem of controlling high degrees of freedom and dynamic multi-body system that is driven by redundant muscle units. A critical challenge in the control perspective of skeletal joints with antagonistic muscle pairs is finding methods robust to address this ill-posed nonlinear problem. To address this computational problem, we implemented a twofold optimization and learning framework to be specialized in addressing the redundancies in the muscle control . In the first part, we used model predictive control to obtain energy efficient skeletal trajectories to mimick human movements. The second part is to use deep reinforcement learning to obtain a sequence of stimulus to be given to muscles in order to obtain the skeletal trajectories with muscle control. We observed that the desired stimulus to muscles is only efficiently constructed by integrating the state and control input in a closed-loop setting as it resembles the proprioceptive integration in the spinal cord circuits. In this work, we showed how a variety of different reference trajectories can be obtained with optimal control and how these reference trajectories are mapped to the musculoskeletal control with deep reinforcement learning. Starting from the characteristics of human arm movement to obstacle avoidance experiment, our simulation results confirm the capabilities of our optimization and learning framework for a variety of dynamic movement trajectories. In summary, the proposed framework is offering a pipeline to complement the lack of experiments to record human motion-capture data as well as study the activation range of muscles to replicate the specific trajectory of interest. Using the trajectories from optimal control as a reference signal for reinforcement learning implementation has allowed us to acquire optimum and human-like behaviour of the musculoskeletal system which provides a framework to study human movement in-silico experiments. The present framework can also allow studying upper-arm rehabilitation with assistive robots given that one can use healthy subject movement recordings as reference to work on the control architecture of assistive robotics in order to compensate behavioural deficiencies. Hence, the framework opens to possibility of replicating or complementing labour-intensive, time-consuming and costly experiments with human subjects in the field of movement studies and digital twin of rehabilitation.

Range-relaxed strategy applied to the Levenberg–Marquardt method with uniformly convex penalization term in Banach spaces

In this paper we propose the employment of the so-called range-relaxed criteria Boiger et al (2020 IMA J. Numer. Anal. 40 606–627) for choosing the regularization parameters (or equivalently, the Lagrange multipliers) of the Levenberg–Marquardt method for solving nonlinear ill-posed problems in Banach spaces. The proposed algorithm employs the Bregman distance induced by a uniformly convex functional and allows the use of a penalization generated from the total variation semi-norm. We present a geometrical interpretation of the method and deliver a complete convergence analysis, including stability and regularization properties. Further, we show that our new method is competitive by testing it with real data in the complete electrode model of 2D electrical impedance tomography.

On the asymptotical regularization with convex constraints for nonlinear ill-posed problems

We investigate the method of asymptotical regularization for solving nonlinear ill-posed problems F(x)=y in Hilbert spaces. A general uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, thus the algorithm can be applied for sparsity and discontinuity reconstruction. Assuming certain conditions concerning the nonlinear operator F and functional Θ, we establish the convergence and stability of the method.

A simplified iteratively regularized projection method for nonlinear ill-posed problems

In this paper, we propose a projection method for the numerical solution of the simplified iteratively regularized Gauss-Newton method of nonlinear integral equations for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive the order optimality. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

Generalized inexact Newton regularization for nonlinear ill-posed problems in Banach spaces

In this paper, we generalize inexact Newton regularization methods to solve nonlinear inverse problems from a reflexive Banach space to a Banach space. The image space is not necessarily reflexive so that the method can be used to deal with various types of noise such as the Gaussian noise and the impulsive noise. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying the regularization technique to the local linearized equations. Under some assumptions, in particular, the reflexivity of the image space is not required, we present a novel convergence analysis of the inexact Newton regularization method with inner scheme defined by Landweber iteration. Furthermore, by employing a two-point gradient method as inner regularization scheme to accelerate the convergence, we propose an accelerated version of inexact Newton–Landweber method and present the detailed convergence analysis. The numerical simulations are provided to demonstrate the effectiveness of the proposed methods in handling different kinds of noise and the fast convergence of the accelerated method.

Multi-exponential model to describe pressure-dependent P- and S-wave velocities and its use to estimate the crack aspect ratio

We present new quantitative model describing the pressure dependence of acoustic P- and S-wave velocities. Assuming that a variety of individual mechanisms or defects (such as cracks, pore collapse and grain crushing) can contribute to the pressure-dependent change of the wave velocity, we order a characteristic pressure to all of them and allow a series of exponential terms in the description of the (P- and S-waves) velocity-pressure function. We estimate the parameters of the multi-exponential rock physical model in inversion procedures using laboratory measured P- and S-wave velocity data. As is known, the conventional damped least squares method gives acceptable results only when one or two individual mechanisms are assumed. Increasing the number of exponential terms leads to highly nonlinear ill-posed inverse problem. Due to this reason, we develop the spectral inversion method (SIM) in which the velocity amplitudes (the spectral lines in the characteristic pressure spectrum) are only considered as unknowns. The characteristic pressures (belonging to the velocity amplitudes) are excluded from the set of inversion unknowns, instead, they are defined in a set of fixed positions equidistantly distributed in the actual interval of the independent variable (pressure). Through this novel linear inversion method, we estimate the parameters of the multi-exponential rock physical model using laboratory measured P- and S-wave velocity data. The characteristic pressures are related to the closing pressures of cracks which are described by well-known rock mechanical relationships depending on the aspect ratio of elliptical cracks. This gives the possibility to estimate the aspect ratios in terms of the characteristic pressures.

Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method

Fluid-saturated porous media are two-phase media, which are composed of solid and liquid phases. Biot theory for fluid-saturated porous media holds that underground media are composed of porous elastic solid and compressible viscous fluid filled with pore space. Compared with the single-phase media theory, the fluid-saturated porous media theory can describe the subsurface media more precisely, and the elastic wave equations in the fluid-saturated porous media contain more parameters used to describe the formation properties. Therefore, fluid-saturated porous media theory is widely used in geophysical exploration, seismic engineering, and other fields.This paper considers the porosity reconstruction problem of the elastic wave equations in the fluid-saturated porous media based on Biot theory, which is a typically nonlinear ill-posed inverse problem from mathematical viewpoint. The proposed method is a novel iteration regularization scheme based on the homotopy perturbation technique. To verify the validity and applicability, numerical experiments of two-dimensional and three-dimensional porosity models have been carried out. Numerical results illustrate that this method can overcome the numerical instability and are robust to data noise in the reconstruction procedure. Furthermore, compared with the classical regularized Gauss-Newton method, the homotopy perturbation method greatly widens the convergence region while keeping the fast convergence rate.

Mixed scale dense convolutional networks for x-ray phase contrast imaging

X-ray in-line phase contrast imaging relies on the measurement of Fresnel diffraction intensity patterns due to the phase shift and the attenuation induced by the object. The recovery of phase and attenuation from one or several diffraction patterns is a nonlinear ill-posed inverse problem. In this work, we propose supervised learning approaches using mixed scale dense (MS-D) convolutional neural networks to simultaneously retrieve the phase and the attenuation from x-ray phase contrast images. This network architecture uses dilated convolutions to capture features at different image scales and densely connects all feature maps. The long range information in images becomes quickly available, and greater receptive field size can be obtained without losing resolution. This network architecture seems to account for the effect of the Fresnel operator very efficiently. We train the networks using simulated data of objects consisting of either homogeneous components, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, or heterogeneous components, consisting of various materials. We also train the networks in the image domain by applying a simple initial reconstruction using the adjoint of the Fréchet derivative. We compare the results obtained with the MS-D network to reconstructions using U-Net, another popular network architecture, as well as to reconstructions using the contrast transfer function method, a direct phase and attenuation retrieval method based on linearization of the direct problem. The networks are evaluated using simulated noisy data as well as images acquired at NanoMAX (MAX IV, Lund, Sweden). In all cases, large improvements of the reconstruction errors are obtained on simulated data compared to the linearized method. Moreover, on experimental data, the networks improve the reconstruction quantitatively, improving the low-frequency behavior and the resolution.

An asymptotical regularization with convex constraints for inverse problems

We investigate the method of asymptotical regularization for the stable approximate solution of nonlinear ill-posed problems F(x) = y in Hilbert spaces. The method consists of two components, an outer Newton iteration and an inner scheme providing increments by solving a local coupling linearized evolution equations. In addition, a non-smooth uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, including L 1-liked and total variation-like penalty terms. We establish convergence properties of the method, derive stability estimates, and perform the convergence rate under the Hölder continuity of the inverse mapping. Furthermore, based on Runge–Kutta (RK) discretization, different kinds of iteration schemes can be developed for numerical realization. In our numerical experiments, four types iterative scheme, including Landweber type, one-stage explicit, implicit Euler and two-stage RK are presented to illustrate the performance of the proposed method.

Image Reconstruction of Conductivity Distribution With Combined L 1-Norm Fidelity and Hybrid Total Variation Penalty

Image reconstruction of conductivity distribution is a highly nonlinear ill-posed inverse problem in electrical impedance tomography (EIT). To solve the problem, Tikhonov regularization and total variation (TV) regularization methods have been generally adopted. However, reconstructed images suffer from excessive smoothness or staircase effect. In this article, a novel approach is proposed for imaging conductivity distribution. Unlike traditional regularization methods, fidelity term based on <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-norm is introduced to stabilize the ill-posed problem and enforce sparsity in the solution. Moreover, a first-order and high-order TV combined hybrid penalty term is introduced to keep the sharp profile and restrain the staircase effect. During image reconstruction, regularization parameter and weighting factor are determined by the adaptive method. In addition, a soft-thresholding operator is introduced to remove artifact and enhance robustness to noise. Numerical simulations have been performed to validate the performance of the proposed approach against Landweber, Tikhonov, TV, high-order TV, and iterated soft-thresholding methods. The impacts of conductivity distribution, conductivity contrast, contact impedance, and boundary shape on reconstruction are studied. Besides, robustness to noise and computation time are evaluated. The performance of the proposed approach is also verified by phantom experiments. Simulation and experimental results demonstrate that the proposed approach performs well in image reconstruction of conductivity distribution. Clearer background is observed and inclusions with sharp or smooth edges can be better reconstructed.