The Landweber iteration applied to inverse conductive scattering problems

When we use a discretization by finite differences, to solve differential equations we find problems at the border of the domain of the solution. If the solution is also immersed in a ill-posed inverse problem; we can find very bad solutions. In this paper we apply a discretization of two - sided difference quotients method to solve Ill-posed inverse self-adjoint elliptic problem [Kirsch(2011)]. Some numerical examples showing the effectiveness of this method and we will use mollification techniques to smooth the solutions.

The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has good generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.

We consider parameter identification problems in parametrized partial differential equations (PDEs). These lead to nonlinear ill-posed inverse problems. One way of solving them is using iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.

Geophysical Inverse Theory and Regularization Problems

We study nonlinear inverse problems of the form F ( x ) = y , and their stable solution via iterative regularization methods, in particular by Newton-type methods, which are well-known for their fast convergence for well-posed problems. A basic step of Newton’s method consists of calculating the update Δ x k via solution of a linearized equation , which will in general be ill-posed if the nonlinear problem is. Thus, for ill-posed problems, the linearized equations have to be solved by some regularization method. In particular for large scale problems, e.g., inverse problems in partial differential equations, where F ( x ) is only defined implicitly via the solution of a PDE, iterative methods have to be used for this purpose. In order to keep the overall effort, i.e., the overall number of iterations, as small as possible, appropriate preconditioning has to be applied. We propose and analyse a general preconditioning strategy in Hilbert scales, and show that the overall number of iterations can be reduced to about the square root by preconditioning. Moreover, in many examples differential operators can be used as preconditioners, and thus preconditioning is almost for free. The theoretical results are illustrated in numerical examples. A comparison with preconditioned Landweber iteration shows that the iteration numbers can be further reduced, if fast iterative methods, e.g., the v -methods, are used for the solution of the linearized problems.

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it.

Remote sensing of ocean color from space aims at retrieving from a noisy top-of-atmosphere radiance the values taken by some relevant quantities like the chlorophyll-a concentration or the marine reflectance. From a mathematical perspective, it is an ill-posed inverse problem with a highly nonlinear operator. Few techniques are available in the case of a nonlinear inverse problem; even its theoretical study is far from easy, yet some techniques may be used in a practical setting when the noise distribution is known. However in the case of ocean color remote sensing, the noise encompasses several types of error owing to the forward operator approximation (radiative transfer model) as well as to calibration and pure measurement noise. Hence the noise distribution is unknown. In this work, a stochastic technique is proposed to first infer a noise distribution, which is next used to retrieve the marine reflectance in a least-square prediction setting by a regression model. The methodology is illustrated on actual data originating from the SeaWiFS sensor.

Distribution of electrical potentials over the surface of the heart, which is called the epicardial potential distribution, is a valuable tool to understand whether there is a defect in the heart. Direct measurement of these potentials requires highly invasive procedures. An alternative is to reconstruct these epicardial potentials non-invasively from the body surface potentials, which constitutes one form of the ill-posed inverse problem of electrocardiography (ECG). The goal of this study is to solve the inverse problem of ECG using several regularization methods and compare their performances. We employed Tikhonov Regularization, Truncated Singular Value Decomposition (TSVD), Least Squares QR (LSQR) methods in this study. We compared the effectiveness of these regularization methods to solve the ill-posed inverse ECG problem. Some of the regularization methods require a regularization parameter to solve the inverse problem. We used the well-known L-Curve method to obtain the regularization parameter. The performance of the regularization methods for solving the inverse ECG problem was also evaluated based on a realistic heart-torso model simulation protocol. In this paper, we also investigated the usage of genetic algorithm (GA) for regularizing the ill-posed inverse ECG problem. The results showed that GA can be applied to regularize the ill-posed problem when combined with the results of conventional regularization methods or additional information about solutions.

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

This reference is for an abstract only. A full paper was not submitted for this conference. Abstract AVO (amplitude-varying-with-offset) is a seismic prestack inversion technique for estimating elastic parameters of the subsurface. The seismic AVO inversion problem can be formulated as a linearized or nonlinear inverse problem. It is a multidimensional and highly ill-posed inverse problem. It is affected by strong noise and measurement uncertainty. Therefore, in seismic AVO analysis, the goal is not only to find the model that best-fits the data, but also to characterize the uncertainty of the analysis. Uncertainty characterization of seismic analysis makes the geophysical interpretation more reliable. The Bayesian formalism generally constitutes a powerful approach for solving many seismic inverse problems. It allows combining available prior knowledge with the information contained in the seismic data. The solution of a Bayesian inverse problem is given by the joint posterior distribution of model parameters. Stochastic approximations of the true posterior are exemplified by the MCMC method and more recently by the particle filtering (PF) method. For most seismic inverse problems, stochastic approximation techniques are computationally unappealing. The maximum a posteriori (MAP) is the simplest Bayesian method for estimating model parameters. It provides the explanation that maximizes the posterior distribution. The maximum likelihood (ML) is a popular non-Bayesian method for estimating model parameters. It provides the explanation that maximizes the likelihood of the data. It assumes that the model parameters are deterministic and omits the pertinent prior knowledge on them. In practice, both MAP and ML point estimators have severe problems with over-fitting the data and model order estimation. In this paper, the recently-developed variational Bayesian learning is proposed as a new method for solving the linearized seismic AVO inversion problem. Variational Bayesian learning allows the true joint posterior distribution of model parameters to be approximated by a simpler approximating ensemble for which the required inferences are tractable. The main advantage in resorting to variational Bayesian learning is its robustness to the over-fitting problem, which is major in practice. This probabilistic machine learning method allows finding an optimum balance between the representation power and the model complexity of the data.

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

This paper is devoted to identification problems in polymer crystallization processes. As a first step the identification of nucleation and growth rates from boundary measurements will be discussed in the case of one-dimensional crystallization. This leads to an identification problem in a coupled system of a hyperbolic and a parabolic equation. An algorithm for the solution of this inverse problem will be developed using Landweber iteration as a regularization method. The behaviour of the algorithm will be studied both numerically and mathematically on the basis of the recently developed convergence theory for iterative regularization methods for nonlinear inverse problems.

Solving a non-linear and high-dimensional inverse problem is challenging in computational science and engineering. Sampling-based methods require a large number of model evaluations; gradient-based methods require fewer model evaluations but only find local minima. Multifidelity optimization combines the low-fidelity model and the high-fidelity model to achieve both high accuracy and efficiency. In this paper, we present a bi-fidelity approach to solving non-linear inverse problems. In the bi-fidelity inversion method, the low-fidelity model is used to acquire a good initial guess, and the high-fidelity model is used to locate the global minimum. Combined with a multi-start optimization scheme, the proposed approach significantly increases the possibility of finding the global minimum for nonlinear inverse problems with many local minima. The method is applied to the inversion of electromagnetic well logging data, which is difficult to solve using traditional gradient-based methods. The bi-fidelity method provides promising inversion results and can be easily applied to traditional gradient-based methods.

Electrical Resistance Tomography (ERT) aims to estimate the electrical properties at the interior of an object from current-voltage measurements on its boundary. Image reconstruction of ERT is a nonlinear and ill-posed inverse problem. Regularization technique is used to solve the inverse problems and two-step iterative shrinkage/thresholding algorithm (TwIST) is a relatively effective method. In order to further improve the noise immunity of iterative threshold algorithm, an improved two-step iterative shrinkage/thresholding algorithm based on 3D modelling is presented in this paper. We transform the sensitivity matrix by singular value decomposition (SVD), and then remove the small singular values which are easy to cause the instability of the solution. Both simulation and experimental results demonstrate that the proposed method can improve the imaging quality obviously.

Magnetic induction tomography (MIT) can be used to probe electrical conductivity variation of an object. It offers a nondestructive and contactless means of imaging the internal conductivity distribution of the object. The forward problem in MIT is a general eddy current problem, which is required to estimate the measured data for a given conductivity distribution. The edge finite-element method with a formulation using a magnetic vector potential has been implemented to solve the forward problem. Conductivity reconstruction in MIT is a nonlinear and ill-posed inverse problem. The regularization techniques are required to incorporate a priori knowledge of the conductivity distribution for a stable solution of this inverse problem. This article presents a conductivity interface reconstruction technique, which has an inherent regularization property. Instead of calculating the conductivity distribution in the whole region of interest, the interface between two different conductivities is reconstructed. A narrowband level-set method has been implemented to describe the interfaces. An iterative optimization scheme has been used to modify the interface estimation in each iteration step, so that the predicated forward solution is closed to the measured data. The results are presented using experimental data representative of an application of MIT in molten metal flow visualization.