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Hybrid approaches combining machine learning with traditional inverse problem solution methods represent a promising direction for the further development of inverse modeling algorithms. The paper proposes an approach to emission source identification from measurement data for advection–diffusion–reaction models. The approach combines general-type source identification and post-processing refinement: first, emission source identification by measurement data is carried out by a sensitivity operator-based algorithm, and then refinement is done by incorporating a priori information about unknown sources. A general-type distributed emission source identified at the first stage is transformed into a localized source consisting of multiple point-wise sources. The second, refinement stage consists of two steps: point-wise source localization and emission rate estimation. Emission source localization is carried out using deep learning with convolutional neural networks. Training samples are generated using a sensitivity operator obtained at the source identification stage. The algorithm was tested in regional remote sensing emission source identification scenarios for the Lake Baikal region and was able to refine the emission source reconstruction results. Hence, the aggregates used in traditional inverse problem solution algorithms can be successfully applied within machine learning frameworks to produce hybrid algorithms. [ABSTRACT FROM AUTHOR]

• An advanced Reference-Constrained Image Restoration algorithm is proposed for solving the image restoration task. • Our algorithm takes advantage of image prior information and merges it with Regularization by Denoising. • By using multiple reference images, we increase the possibility to improve the quality of the reconstructed images. • The achievable performance of the proposed algorithm outperforms Regularization by Denoising. Image restoration is always a long-term problem in low-level computer vision, which has many practical applications. In this paper, we devote to reconstructing a single image from compressive measurements with the assistance of one or multiple reference images, by using the similarity prior information of them. More specifically, we propose to bring in the concept of Regularization by Denoising, which leverages existing denoising engine to regularize inverse problems, and illustrate how the reference images and this powerful algorithm can be merged into a new highly effective recovery approach, which is named Reference Image Constrained Regularization by Denoising for solving various image restoration tasks. Since the similarity between the reference and the ground truth is not guaranteed, or the degree of the similarity is different, we extend the proposed algorithm by adding adaptive weighting matrix to the difference between the target image and the reference images to further improve the performance. We evaluate experimentally the proposed algorithm in the image deblurring and single image super-resolution problems. Furthermore, we also carry out our method on video super-resolution task, in the case of multiple reference images available. We compare the advanced reference-constrained image restoration algorithm with previous methods about reconstruction accuracy and computation cost. The experimental results show that the proposed method outperforms state-of-the-art algorithms, including Regularization by Denoising, in terms of both the Peak Signal to Noise Ratio and the visual performance. [ABSTRACT FROM AUTHOR]

This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties. [ABSTRACT FROM AUTHOR]

In this paper, we bring forward a distributed algorithm for the assignment of a prescribed Laplacian spectrum for path graphs by means of asymmetric weight assignment. We first describe the meaningfulness and the relevance of this mathematical setting in modern technological applications, and some examples are reported, revealing its practical usefulness in a variety of applications. Then, the solution is derived both theoretically and through an algorithm. Special attention is devoted to a distributed implementation of the main algorithm, which is a valuable feature for several modern applications. Finally, the positivity is discussed, which is revealed to be a consequence of the strict interlacing property. [ABSTRACT FROM AUTHOR]

An iterative neural network framework is proposed in this paper for the human-induced Ground Reaction Forces (GRF) replication with an inertial electrodynamic mass actuator (APS 400). This is the first approach to the systematization of dynamic load tests on structures on a purely objective, repeatable and pedestrian-independent basis. Therefore, an inversion-free offline algorithm based on Machine Learning techniques has been applied for the first time on an electrodynamic shaker, without requiring its inverse model to tackle the inverse problem of successful force reconstruction. The proposed approach aims to obtain the optimal drive signal to minimize the error between the experimental shaker output and the reference force signal, measured with a pair of instrumented insoles (Loadsol©) for human bouncing at different frequencies and amplitudes. The optimal performance, stability and convergence of the system are verified through experimental tests, achieving excellent results in both time and frequency domain. [ABSTRACT FROM AUTHOR]

Fluence map optimization for intensity-modulated radiation therapy planning can be formulated as a large-scale inverse problem with competing objectives and constraints associated with the tumors and organs at risk. Unfortunately, clinically relevant dose–volume constraints are nonconvex, so standard algorithms for convex problems cannot be directly applied. Although prior work focuses on convex approximations for these constraints, we propose a novel relaxation approach to handle nonconvex dose–volume constraints. We develop efficient, provably convergent algorithms based on partial minimization, and show how to adapt them to handle maximum-dose constraints and infeasible problems. We demonstrate our approach using the CORT data set and show that it is easily adaptable to radiation treatment planning with dose–volume constraints for multiple tumors and organs at risk. Summary of Contribution: This paper proposes a novel approach to deal with dose–volume constraints in radiation treatment planning optimization, which is inherently nonconvex, mixed-integer programming. The authors tackle this NP-hard problem using auxiliary variables and continuous optimization while preserving the problem's nonconvexity. Algorithms to efficiently solve the nonconvex optimization problem presented in this paper yield computation speeds suitable for a busy clinical setting. [ABSTRACT FROM AUTHOR]

Nonconvex constraints are valuable regularizers in many optimization problems. In particular, sparsity constraints have had a significant impact on sampling theory, where they are used in compressed sensing and allow structured signals to be sampled far below the rate traditionally prescribed. Nearly, all of the theory developed for compressed sensing signal recovery assumes that samples are taken using linear measurements. In this paper, we instead address the compressed sensing recovery problem in a setting where the observations are nonlinear. We show that, under conditions similar to those required in the linear setting, the iterative hard thresholding algorithm can be used to accurately recover sparse or structured signals from few nonlinear observations. Similar ideas can also be developed in a more general nonlinear optimization framework. In the second part of this paper, we therefore present related result that shows how this can be done under sparsity and union of subspaces constraints, whenever a generalization of the restricted isometry property traditionally imposed on the compressed sensing system holds. [ABSTRACT FROM PUBLISHER]

One of the key tools in an organization's performance management is the goal tree, which is used for solving both direct and inverse problems. This research deals with goal setting based on a model of the future by presenting the goal and subgoal in the form of concrete quantitative and qualitative characteristics and stepwise formation of factors. A stepwise solution to a factor generation problem is considered on the basis of mathematical symmetry. This paper displays an algorithm for solving hierarchical inverse problems with constraints, which is based on recursively traversing the vertices that constitute the separate characteristics. Iterative methods, modified for the case of nonlinear models and the calculation of constraints, were used to generate solutions to the subproblems. To realize the algorithm, the object-oriented architecture, which simplifies the creation and modification of software, was elaborated. Computational experiments with five types of models were conducted, and the solution to a problem related to fast-food restaurant profit generation was reviewed. The metrics of remoteness from set values and t-statistics were calculated for the purpose of testing the received results, and solutions to the subproblems, with the help of a mathematical package using optimization models and a method of inverse calculations, were also provided. The results of computational experiments speak to the compliance of the received results with set constraints and the solution of separate subproblems with the usage of the mathematical package. The cases with the highest solution accuracy reached are specified. [ABSTRACT FROM AUTHOR]

As a key technology for tight gas stimulation, refracturing plays an important role in tight gas development. In the production process of tight gas wells, the reservoir or fracturing process may cause the hydraulic fractures to gradually fail and the production to continuously decrease. In order to restore the productivity of a single well, it is necessary to refract the well to reopen the failed fractures or fracturing. Reasonable refracturing timing and optimization of refract fracture parameters are important guarantees to ensure the benefits of refracturing in tight gas wells, and relevant research on it can provide theoretical and technical guidance for field construction design. Based on the inverse problem of the dynamic prediction model of tight gas well productivity, this paper proposes an inversion method of formation and fracture parameters before refracturing based on Bayesian inversion algorithm. Finally, based on the geology and development data of the fractured wells in the Sulige gas field, the field application of refracting well selection, determination of refracting reasonable timing, and prediction of refracting effect is carried out. The actual production data are compared, and it is shown that this method can provide theoretical guidance for high-efficiency production-increasing construction on-site. [ABSTRACT FROM AUTHOR]

We study Sturm–Liouville differential operators on the time scales consisting of finitely many isolated points and closed intervals. In the author's previous paper, it was established that such operators are uniquely determined by the spectral characteristics of all classical types. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm–Liouville boundary-value problems on time scales with one common boundary condition alternate. [ABSTRACT FROM AUTHOR]

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L 2. We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows. • Introduce a model to find desired coefficient. • No need an initial guess. • Fast convergence. • Interesting numerical results. [ABSTRACT FROM AUTHOR]

Damage detection through an inverse optimization problem has been investigated by many researchers. Recently, Modal Strain Energy (MSE) has been utilized as an index (MSEBI) for damage localization that serves to guide the optimization. This guided approach considerably reduces the computational cost and increases the accuracy of optimization. Although this index mostly exhibits an acceptable performance, it fails to find some damaged elements' locations in some cases. The aim of this paper is twofold. Firstly, a Graph-based Adaptive Threshold (GAT) is proposed to identify some of those elements that are not detected by basic MSEBI. GAT relies on the concepts from graph theory and MSE working as a simple anomaly detection technique. Secondly, an Improved version of the Water Strider Algorithm (IWSA) is introduced, applied to the damage detection problems with incomplete modal data and noisecontaminated inputs. Several optimization algorithms, including the newly-established Water Strider Algorithm (WSA), are utilized to test the proposed method. The investigations on several damage detection problems demonstrate the GAT and IWSA's satisfactory performance compared to the previous methods. [ABSTRACT FROM AUTHOR]

In this paper, the inverse problem of the determination of an unknown discontinuous parameter in elliptic equations is considered. A constrained optimization problem is formulated using the TV regularization and a specified selection term, in order to absorb the effect of noise and singularities. The discretization scheme used in this work is a combined finite element/volume method, then we establish the discrete version of optimization problem. The convergence and stability analysis are performed in term of noise level and diameter of singular set. We have considered the projected Wolfe algorithm to design our proposed minimization method. At the end we show some numerical experiments to prove the validity of theoretical results and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a modified meta-heuristic algorithm based on the blind naked mole-rat (BNMR) algorithm to solve the multiple standard benchmark problems. We then apply the proposed algorithm to solve an engineering inverse problem in the electromagnetic field to validate the results. The main objective is to modify the BNMR algorithm by employing two different types of distribution processes to improve the search strategy. Furthermore, we proposed an improvement scheme for the objective function and we have changed some parameters in the implementation of the BNMR algorithm. The performance of the BNMR algorithm was improved by introducing several new parameters to find the better target resources in the implementation of a modified BNMR algorithm. The results demonstrate that the changed candidate solutions fall into the neighborhood of the real solution. The results show the superiority of the propose method over other methods in solving various mathematical and electromagnetic problems. [ABSTRACT FROM AUTHOR]

In this paper we study the inverse problem for Sturm–Liouville operator with periodic condition and a finite number discontinuities. We give the uniqueness theorem by two spectra and the sign sets, and the reconstruction algorithm. [ABSTRACT FROM AUTHOR]

Image acquisition in many biomedical imaging modalities is corrupted by Poisson noise followed by additive Gaussian noise. While total variation and related regularization methods for solving biomedical inverse problems are known to yield high quality reconstructions in most situations, such methods mostly use log-likelihood of either Gaussian or Poisson noise models, and rarely use mixed Poisson-Gaussian (PG) noise model. There is a recent work which deals with exact PG likelihood and total variation regularization. This method adapts the primal-dual approach involving gradients steps on the PG log-likelihood, with step size limited by the inverse of its Lipschitz constant. This leads to limitations in the convergence speed. Although the alternating direction method of multipliers (ADMM) algorithm does not have such step size restrictions, ADMM has never been applied for this problem, for the possible reason that PG log-likelihood is quite complex. In this paper, we develop an ADMM based optimization for roughness minimizing image restoration under PG log-likelihood. We achieve this by first developing a novel iterative method for computing the proximal solution of PG log-likelihood, deriving the termination conditions for this iterative method, and then integrating into a provably convergent ADMM scheme. We experimentally demonstrate that the proposed method outperform primal-dual method in most of the cases. [ABSTRACT FROM AUTHOR]

Regularization is necessary for solving non-linear ill-posed inverse problems arising in different fields of geosciences. The base of a suitable regularization is the prior expressed by the regularizer, which can be non-adaptive or adaptive (data-driven), smooth or non-smooth, variational-based or not. Nevertheless, tailoring a suitable and easy-to-implement prior for describing geophysical models is a non-trivial task. In this paper, we propose two generic optimization algorithms to implement arbitrary regularization in non-linear inverse problems such as full-waveform inversion (FWI), where the regularization task is recast as a denoising problem. We assess these optimization algorithms with the plug-and-play block matching (BM3D) regularization algorithm, which determines empirical priors adaptively without any optimization formulation. The non-linear inverse problem is solved with a proximal Newton method, which generalizes the traditional Newton step in such a way to involve the gradients/subgradients of a (possibly non-differentiable) regularization function through operator splitting and proximal mappings. Furthermore, it requires to account for the Hessian matrix in the regularized least-squares optimization problem. We propose two different splitting algorithms for this task. In the first, we compute the Newton search direction with an iterative method based upon the first-order generalized iterative shrinkage-thresholding algorithm (ISTA), and hence Newton-ISTA (NISTA). The iterations require only Hessian-vector products to compute the gradient step of the quadratic approximation of the non-linear objective function. The second relies on the alternating direction method of multipliers (ADMM), and hence Newton-ADMM (NADMM), where the least-squares optimization subproblem and the regularization subproblem in the composite objective function are decoupled through auxiliary variable and solved in an alternating mode. The least-squares subproblem can be solved with exact, inexact, or quasi-Newton methods. We compare NISTA and NADMM numerically by solving FWI with BM3D regularization. The tests show promising results obtained by both algorithms. However, NADMM shows a faster convergence rate than NISTA when using L-BFGS to solve the Newton system. [ABSTRACT FROM AUTHOR]

Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA. [ABSTRACT FROM AUTHOR]

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its ${M}+1$ consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is $1/{M}$ and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than $1/{M}$ apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like $\text {SRF}^{2\lambda -2} \times \text {Noise}$ , where the Super-Resolution Factor (SRF) governs the difficulty of the problem and $\lambda $ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of $M$ in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments. [ABSTRACT FROM AUTHOR]

Registration refers to the useful process of aligning two similar but different intensity image functions in order to either track changes or combine information. Variational models are capable of finding transform maps containing large and non-uniform deformations between such a pair of images. Since finding a transform map is an inverse problem, as with all models, suitable regularisation is necessary to overcome the non-uniqueness of the problem. In the case of diffusion type models regularisation terms impose smoothness on the transformation by minimising the gradient of the flow field. The diffusion model also coincides with the basic model for optical flow frameworks of Horn and Schunck (1981). The biggest drawback with variational models is the large computational cost required to solve the highly non-linear system of PDEs; Chumchob and Chen (2011) developed a non-linear multigrid (NMG) method to address this cost problem. However, a closer look at the analysis of the NMG scheme highlighted omissions which affected the convergence of the NMG scheme. Moreover, the NMG method proposed by Chumchob and Chen did not impose any control of non-physical folding which invalidates a map. This paper has proposed several key ideas. First we re-evaluate the analysis of the NMG method to show how the omissions in Chumchob and Chen (2011) have a noticeable impact on the convergence of the NMG method. In addition, we also provide a way of estimating the convergence rate of a solver on the coarsest grid in order to estimate the number of iterations that will be required to obtain a solution with appropriate accuracy. Second we propose an extension to the Chumchob–Chen NMG method which controls any folding within the deformation. Experimental results on the proposed multigrid framework demonstrate improvements in convergence and the accuracy of registrations compared with previous methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial accelerated iterative algorithm for approximating a common solution of a minimization problem and a fixed point problem for quasi-pseudocontractive mapping in a real Hilbert space. Using the algorithm, we prove a strong convergence theorem for approximating a common solution of a minimization problem and a fixed point problem for quasi-pseudocontractive mapping. Furthermore, we give an application of our main result to solve convexly constrained linear inverse problems, and we also present a numerical example of our algorithm to illustrate its applicability. [ABSTRACT FROM AUTHOR]

In previous papers, the authors introduced and characterized a class of matrices called { K , s + 1 } -potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying K A s + 1 K = A for a given matrix A ∈ C n × n and a given natural number s . The cases s = 0 and s ≥ 1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods. [ABSTRACT FROM AUTHOR]

This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed–Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp–Massey algorithm. Two additional algorithms are obtained as easy variations of the basic algorithm: the first variation is entirely new, while the second variation may be viewed as a version of the Euclidean algorithm. Decoding Reed–Solomon codes (and some related codes) can be reduced to the partial-inverse problem, both via the standard key equation and, more naturally, via an alternative key equation with a new converse. Shortened and singly-extended Reed–Solomon codes are automatically included. Using the properties of the partial-inverse problem, two further key equations with attractive properties are obtained. The paper also points out a variety of options for interpolation. [ABSTRACT FROM PUBLISHER]

In case of a release of a hazardous material (e.g., a chemical or a biological agent) in the atmosphere, estimation of the source from concentration observations (provided by a network of sensors) is a challenging inverse problem known as the atmospheric source term estimation (STE) problem. This study emphasizes a method, known in the literature as the renormalization inversion technique, for addressing this problem. This method provides a solution that has been interpreted as a weighted minimal norm solution and can be computed in terms of a generalized inverse of the sensitivity matrix of the sensors. This inverse is constructed by using an appropriate diagonal weight matrix whose components fulfill the so-called renormalizing conditions. The main contribution of this paper is that it proposes a new compact algorithm (it requires less than 15 lines of MATLAB code) to obtain, in a fast and efficient way, those optimal weights. To show that the algorithm, based on the properties of the resolution matrix, matches the requirements of emergency situations, analysis of the computational complexity and memory requirements is included. Some numerical experiments are also reported to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a characteristic feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that 1) sidesteps the technical challenges present in previous work, 2) admits estimators that can readily be computed using off-the-shelf LASSO algorithms, and 3) hints at a general framework for broad classes of noise in sparse linear inverse problems. At the heart of this new approach lies a weighted LASSO estimator for which data-dependent weights are based on Poisson concentration inequalities. Unlike previous analyses of the weighted LASSO, the proposed analysis depends on conditions which can be checked or shown to hold in general settings with high probability. 2000 Math Subject Classification: 60E15, 62G05, 62G08, and 94A12. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, we devote our effort to solve a nonlinear inverse problem for identifying a time-dependent convection coefficient in a time-fractional diffusion equation from the measured data at an interior point for one-dimensional case. We prove the existence, uniqueness and regularity of solution for the direct problem by using the fixed point theorem. The stability of inverse convection coefficient problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall's inequalities. We use a modified optimal perturbation regularization algorithm to solve the inverse convection coefficient problem. Two numerical examples are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Phase retrieval is an inverse problem which consists in recovering an unknown signal from a set of absolute squared projections. Recently, gradient descent algorithms have been developed to solve this problem. However, their optimization cost functions are non-convex and non-smooth. To address the non-smoothness of the cost function, some of these methods use truncation thresholds to calculate a truncated step gradient direction. But, the truncation requires designing parameters to obtain a desired performance in the phase recovery, which drastically modifies the search direction update, increasing the sampling complexity. Therefore, this paper develops the Phase Retrieval Smoothing Conjugate Gradient method (PR-SCG) which uses a smoothing function to retrieve the signal. PR-SCG is based on the smooth-ing projected gradient method which is useful for non-convex optimization problems. PR-SCG uses a nonlinear conjugate gradient of the smoothing function as the search direction to accelerate the convergence. Furthermore, the incremental Stochastic Smoothing Phase Retrieval algorithm (SSPR) is developed. SSPR involves a single equation per iteration which results in a simple, scalable, and fast approach useful when the size of the signal is large. Also, it is shown that SSPR converges linearly to the true signal, up to a global unimodular constant. Additionally, the proposed methods do not require truncation parameters. Simulation results are provided to validate the efficiency of PR-SCG and SSPR compared to existing phase retrieval algorithms. It is shown that PR-SCG and SSPR are able to reduce the number of measurements and iterations to recover the phase, compared with recently developed algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional inverse medium problem from noisy observation data. We propose effective reconstruction algorithms to detect the num-ber, the location and the size of the piecewise constant medium within a body, and then we try to recover the unknown shape of inhomogeneous media. This problem is nonlinear and ill-posed, thus we should consider stable and elegant approaches in order to improve the corresponding approximation. We give several examples to show the viability of our proposed algorithms. [ABSTRACT FROM AUTHOR]

In the context of the phase retrieval problem, it is known that certain natural classes of measurements, such as Fourier measurements and random Bernoulli measurements, do not lead to the unique reconstruction of all possible signals, even in combination with certain practically feasible random masks. To avoid this difficulty, the analysis is often restricted to measurement ensembles (or masks) that satisfy a small-ball probability condition, in order to ensure that the reconstruction is unique. This paper shows a complementary result: for random Bernoulli measurements, there is still a large class of signals that can be reconstructed uniquely, namely, those signals that are non-peaky. In fact, this result is much more general: it holds for random measurements sampled from any subgaussian distribution \mathcal D , without any small-ball conditions. This is demonstrated in two ways: 1) a proof of stability and uniqueness and 2) a uniform recovery guarantee for the PhaseLift algorithm. In all of these cases, the number of measurements $m$ approaches the information-theoretic lower bound. Finally, for random Bernoulli measurements with erasures, it is shown that PhaseLift achieves uniform recovery of all signals (including peaky ones). [ABSTRACT FROM PUBLISHER]

In this paper, we combine the fading regularization method with the method of fundamental solutions (MFS) and investigate its application to the Cauchy problem for the two-dimensional Helmholtz equation. We present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of overprescribed noisy data taken on the remaining accessible boundary part for both smooth and piecewise smooth two-dimensional geometries. The accuracy, convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data, are validated by three numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study affine transformations on nilmanifolds and compact abelian groups. For these systems, we show that an equivalent condition to have zero entropy is that the orbit closure of each point has a nice structure. To be precise, the affine systems on those spaces have zero entropy if and only if the orbit closure of each point is isomorphic to an inverse limit of nilsystems. [ABSTRACT FROM AUTHOR]

Abstract: Owing to providing a novel insight for signal and image processing, compressed sensing (CS) has attracted increasing attention. The accuracy of the reconstruction algorithms plays an important role in real applications of the CS theory. In this paper, a generalized reconstruction model that simultaneously considers the inaccuracies on the measurement matrix and the measurement data is proposed for CS reconstruction. A generalized objective functional, which integrates the advantages of the least squares (LSs) estimation and the combinational M-estimation, is proposed. An iterative scheme that integrates the merits of the homotopy method and the artificial physics optimization (APO) algorithm is developed for solving the proposed objective functional. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. For the cases simulated in this paper, the reconstruction accuracy is improved, which indicates that the proposed algorithm is successful in solving CS inverse problems. [Copyright &y& Elsevier]

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting. [ABSTRACT FROM AUTHOR]

This paper presents a new concept for atmospheric reentry online optimal guidance and control using a method called MARE G&C that exploits the different time scale featured by reentry dynamics. The new technique reaches a quasi-analytical solution and simplified computations, even considering both lift-to-drag ratio and aerodynamic roll as control variables; in addition, the paper offers a solution for the challenging path constraints issue, getting inspiration from the inverse problem methodology. The final resulting algorithm seems suitable for onboard predictive guidance, a new need for future space missions. [ABSTRACT FROM AUTHOR]

Background: In diffuse optical tomography (DOT), the image reconstruction is often an ill-posed inverse problem, which is even more severe for breast DOT since there are considerably increasing unknowns to reconstruct with regard to the achievable number of measurements. One common way to address this ill-posedness is to introduce various regularization methods. There has been extensive research regarding constructing and optimizing objective functions. However, although these algorithms dramatically improved reconstruction images, few of them have designed an essentially differentiable objective function whose full gradient is easy to obtain to accelerate the optimization process.Methods: This paper introduces a new kind of non-negative prior information, designing differentiable objective functions for cases of L1-norm, Lp (0 < p < 1)-norm and L0-norm. Incorporating this non-negative prior information, it is easy to obtain the gradient of these differentiable objective functions, which is useful to guide the optimization process.Results: Performance analyses are conducted using both numerical and phantom experiments. In terms of spatial resolution, quantitativeness, gray resolution and execution time, the proposed methods perform better than the conventional regularization methods without this non-negative prior information.Conclusions: The proposed methods improves the reconstruction images using the introduced non-negative prior information. Furthermore, the non-negative constraint facilitates the gradient computation, accelerating the minimization of the objective functions. [ABSTRACT FROM AUTHOR]

Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filtering of nonlinear (nonphysical) Fourier transform data specific to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass filter to include higher frequency components in the reconstruction, in particular, features that are known with high confidence. This information is additionally included in the system of D-bar equations with an independent regularization parameter from that of the extended scattering transform. In this paper, this approach is used in the 2-D D-bar method for admittivity (conductivity as well as permittivity) EIT imaging. Noise-robust reconstructions are presented for simulated EIT data on chest-shaped phantoms with a simulated pneumothorax and pleural effusion. No assumption of the pathology is used in the construction of the prior, yet the method still produces significant enhancements of the underlying pathology (pneumothorax or pleural effusion) even in the presence of strong noise. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the inverse problem on determining the spatial component of the source term in the hyperbolic equation with a time-dependent principal part. Based on a Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. Numerical examples up to three spatial dimensions are presented to demonstrate the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (\mmb f^\star) from Poisson data (\mmb y) cannot be effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of \mmb f^\star from \mmb y in an inverse problem setting, where the number of unknowns may potentially be larger than the number of observations and \mmb f^\star admits sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to \mmb \ell1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. DeMol (2004) . This generalization is useful for solving many practical problems in which more than one constraint are involved. In this regard, we will conclude the findings of many papers (most of which are on image processing) from this generalization. It is hoped that the approach proposed in this paper will be a suitable reference for some applied works where multi-frames, multi-wavelets, or multi-constraints are present in linear inverse problems. [Copyright &y& Elsevier]

Abstract: This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor–corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs. [Copyright &y& Elsevier]

Electrical impedance tomography (EIT) is an imaging modality of determining imaging the interior conductivity distribution within an object. In the paper, generalized back projection algorithm (GBPA) is proposed for EIT image reconstruction. The algorithm is based on node back projection algorithm (NBPA), and extends projection range to arbitrary part of the object. To evaluate the proposed method, GBPA is first applied to image reconstruction using a simulated cylinder model. Compared with conventional back projection algorithm (BPA) and NBPA, the images reconstructed by GBPA are more accurate in terms of several metrics used in the paper. In addition, we present image reconstruction results using a 128-channel EIT system to demonstrate the performance of GBPA in real experiments. [ABSTRACT FROM PUBLISHER]

This paper presents a new neural network (NN) algorithm for real-time retrievals of low amounts of precipitable water vapor (PWV) and integrated liquid water from millimeter-wave ground-based observations. Measurements are collected by the 183.3-GHz G-band vapor radiometer (GVR) operating at the Atmospheric Radiation Measurement (ARM) Program Climate Research Facility, Barrow, AK. The NN provides the means to explore the nonlinear regime of the measurements and investigate the physical boundaries of the operability of the instrument. A methodology to compute individual error bars associated with the NN output is developed, and a detailed error analysis of the network output is provided. Through the error analysis, it is possible to isolate several components contributing to the overall retrieval errors and to analyze the dependence of the errors on the inputs. The network outputs and associated errors are then compared with results from a physical retrieval and with the ARM two-channel microwave radiometer (MWR) statistical retrieval. When the NN is trained with a seasonal training data set, the retrievals of water vapor yield results that are comparable to those obtained from a traditional physical retrieval, with a retrieval error percentage of ~5% when the PWV is between 2 and l0 mm, but with the advantages that the NN algorithm does not require vertical profiles of temperature and humidity as input and is significantly faster computationally. Liquid water path (LWP) retrievals from the NN have a significantly improved clear-sky bias (mean of ~2.4 g/m²) and a retrieval error varying from 1 to about 10 g/ms when the PWV amount is between 1 and 10 mm. As an independent validation of the LWP retrieval, the Iongwave downwelling surface flux was computed and compared with observations. The comparison shows a significant improvement with respect to the MWR statistical retrievals, particularly for LWP amounts of less than 60 g/m². This paper shows that the GVR alone can provide overall improved PWV and LWP retrievals when the PWV amount is less than 10 mm, and, when combined with the MWR, can provide improved retrievals over the whole water-vapor range. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies an inverse problem of identifying the coefficient of parabolic equation when the final observation is given, which has important application in a large fields of applied science. Based on the optimal control framework, the existence and necessary condition of the minimum for the control functional are established. Since the optimal control problem is nonconvex, one may not expect a unique solution. However, in this paper the solution is proved to be locally unique. After the necessary condition is transformed into an elliptic bilateral variational inequality, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well. [Copyright &y& Elsevier]

Inhomogeneous current distribution in lithium-ion cells is a major reason for degradation. A method to measure the locally resolved current distribution is to map the magnetic field caused by the current flowing in the cell. This paper presents an unshielded measurement setup using anisotropic magnetoresistive (AMR) sensors to map the magnetic field of a cell during operation. Moreover, it proposes an algorithm to calculate the two-dimensional current distribution from magnetic field data. The achieved accuracy of the current density is 227 mA cm−2 at a local resolution of 4 mm2. The mathematical problem is reduced to a single matrix multiplication and takes into account specific characteristics of the measurement setup. Simulated and measured data are used to evaluate the proposed algorithm. Two cells with different geometries, capacities and tab designs are tested and compared. The paper provides guidance for the interpretation of the measurement data and introduces a measure to calculate the validity and accuracy, which can also be used to trim the parameters for the inverse problem. Possible fields of applications include end of line tests or state of health investigations where non-destructive and in-situ measurements are needed. [Display omitted] • Nondestructive, locally resolved current distribution measurement. • Setup to precisely measure the magnetic field above the cell. • Algorithm to reconstitute current distribution from magnetic field image. • Current distribution measured for cells with different geometries. • Enables inhomogeneity analysis and defect detection in cells. [ABSTRACT FROM AUTHOR]

Abstract: Parameter estimation problems and heat source/flux reconstruction problems are some of the most frequently encountered inverse heat transfer problems. These problems find their application in many areas of science and engineering. The primary focus of this paper is on the heat transfer parameter estimation for a two-dimensional unsteady heat conduction problem with (a) convection boundary condition and (b) convection and radiation boundary condition. The paper demonstrates the effect of a priori model on the performance of the algorithm at different noise levels in the measured data. The inverse problem is solved using three different a priori models namely normal, log normal and uniform. The posterior PDF is sampled using the Metropolis–Hastings sampling algorithm. Both single-parameter estimation and multi-parameter estimation problems are addressed and the effects of corresponding a priori models are studied. It was found that the mean and maximum a posteriori estimates for thermal conductivity and the convection heat transfer coefficient were insensitive to the a priori model at all the considered noise levels for the single-parameter estimation problem. At high noise levels in the two-parameter estimation problem, the estimates for thermal conductivity and convection coefficient were sensitive to the a priori model. It was also found that the standard deviation of the samples was correlated to the error in estimation in the single-parameter estimation case. In three parameter estimation case, alternate solutions to the same problem were retrieved due to a strong correlation between the convection coefficient and the emissivity. However, a more informative a priori model could address this issue. [Copyright &y& Elsevier]