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The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton). [ABSTRACT FROM AUTHOR]

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]

In this paper, we investigate the application of the fading regularization method with the method of fundamental solutions (MFS) to the ill-posed Cauchy-Stokes problem. For both smooth and piecewise smooth two-dimensional geometries, we present a numerical reconstruction of the missing velocity and normal stress tensor on an inaccessible part of the boundary based on knowledge of over-prescribed noisy data acquired on the remaining accessible boundary part. Three numerical examples demonstrate the proposed numerical algorithm's accuracy, convergence, stability, and efficiency, as well as its ability to deblur the data. [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]

Accelerated regularization algorithms for ill-posed problems have received much attention from researchers of inverse problems since the 1980s. The current optimal theoretical results indicate that some regularization algorithms, e.g. the ν -method and the Nesterov method, are such that under conventional source conditions the optimal convergence rates can be obtained with approximately the square root of the iterations of those needed for the benchmark (i.e. the Landweber iteration). In this paper, we propose a new class of regularization algorithms with parameter n, called the Acceleration Regularization of order n (AR n ). Theoretically, we prove that, for an arbitrary number n > - 1 , AR n can attach the optimal convergence rates with approximately the n + 1 root of the iterations needed for the benchmark method. Moreover, unlike the existing accelerated regularization algorithms, AR n s have no saturation restriction. Some symplectic iterative regularizing algorithms are developed for the numerical realization of AR n . Finally, numerical experiments with integral equations and inverse problems in partial differential equations demonstrate that, at least for n ≤ 2 , the numerical behavior of AR n matches our theoretical findings, also breaking the practical acceleration capability of all existing regularization algorithms. [ABSTRACT FROM AUTHOR]

Gaussian process regression has proven very powerful in statistics, machine learning and inverse problems. A crucial aspect of the success of this methodology, in a wide range of applications to complex and real-world problems, is hierarchical modeling and learning of hyperparameters. The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature. Analysis of their consistency in the large data limit, as well as explicit identification of their implicit bias in parameter learning, are established in this paper for a Matérn-like model on the torus. A particular technical challenge we overcome is the learning of the regularity parameter in the Matérn-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Matérn-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach. [ABSTRACT FROM AUTHOR]

Different geophysical methods use different model parameterizations and inversion algorithms. Thus, combining these different inversion systems and yet adding the nonlinear cross‐gradient constraint in a joint inversion framework might be a big challenge, for instance, as explained further by Moorkamp et al. in 2011, there is a complex interaction between the data misfit terms, regularization and cross‐gradient terms and an imperfect fit to the data is expected. In this paper, we use a sequential algorithm for a two‐dimensional joint inversion of gravity and magnetic data, which tries to avoid these issues by decoupling the gravity inversion, the magnetic inversion and the cross‐gradient minimization processes. The efficiency of the algorithm and developed code is demonstrated by the joint inversion of noisy synthetic data. The results show a significant improvement in the respective models obtained by introducing the cross‐gradient joint inversion over the models obtained by separate inversions for synthetic data and then for field data targeting potash ore source in the AjiChai salt deposit in north‐western Iran. In this application case, the lower density of salt minerals such as potash, compared to its surrounding sedimentary sequences, motivates a gravity study. Furthermore, the relative lower susceptibility of these salt minerals, alongside their diamagnetic effect, makes them a suitable target for magnetic surveys. Separate gravity and magnetic studies had been performed over the deposit; however, a constitutive relationship between density and magnetization within the area of interest supporting a joint inversion had not been established. In this paper, we apply the sequential cross‐gradient approach to perform the first full joint inversion for the AjiChai salt deposit. The magnetic inversion here is performed to recover the magnetization amplitudes rather than the magnetization vector. In fact, we assume there is no remanent magnetization and, therefore, that the magnetization vector is constant and parallel to the geomagnetic field direction. The constructed density and magnetization models are of high concordance with available geological information and previous studies including drilling results. In addition, unlike previous separate inversion models, the models are structurally and geometrically similar. [ABSTRACT FROM AUTHOR]

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]

The "fast iterative shrinkage-thresholding algorithm,"" a.k.a. FISTA, is one of the most well known first-order optimization scheme in the literature, as it achieves the worst-case O(1/k2) optimal convergence rate for objective function value. However, despite such an optimal theoretical convergence rate, in practice the (local) oscillatory behavior of FISTA often damps its efficiency. Over the past years, various efforts have been made in the literature to improve the practical performance of FISTA, such as monotone FISTA, restarting FISTA, and backtracking strategies. In this paper, we propose a simple yet effective modification to the original FISTA scheme which has two advantages: It allows us to (1) prove the convergence of generated sequence and (2) design a so-called lazy-start strategy which can be up to an order faster than the original scheme. Moreover, we propose novel adaptive and greedy strategies which further improve the practical performance. The advantages of the proposed schemes are tested through problems arising from inverse problems, machine learning, and signal/image processing. [ABSTRACT FROM AUTHOR]

• A versatile framework is established for solving reactor inverse problems. • Global optimization algorithms enhance the framework's problem-solving capabilities in previously challenging complex nonlinear scenarios. • A 24-dimension problem concluding the positions of control rods and other parameters is successfully solved. • Tests are based on the real design of the Daya Bay Nuclear Power Plant. In reactor-related scenarios, inverse methods utilizing monitoring data are frequently employed to address issues such as state estimation, accident analysis, and power reconfiguration. These problem-solving approaches can entail converting the original problem into a parameter estimation problem. This paper presents a versatile and scalable framework for solving inverse problems by iterative optimization. To address intricate core parameter problems effectively, we incorporate the CPACT program, a physical thermal engineering solution program. Additionally, we employ the global optimization algorithm known as CSA (Characteristic Statistic Algorithm) to tackle specific challenges associated with nonlinear global optimization. To evaluate the performance of our framework, a numerical simulation model based on the Daya Bay Nuclear Power Plant was used as the test object design as our test subject. [ABSTRACT FROM AUTHOR]

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohnâ€"Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving the inverse problem of restoring the hydrometeor distribution function from radar measurements. We use the Tikhonov regularization algorithm based on the minimization of the residual functional by the conjugate gradient method. [ABSTRACT FROM AUTHOR]

This paper presents an efficient implementation of the inversion algorithm based on the conjugate-gradient method (CGM) for solving inverse scattering problems. The original CGM provides good image reconstruction and robustness to noise-corrupted data. This method requires the solution of the forward scattering problem and the Fréchet derivative operator of the cost function at each iteration step. However, these procedures can make the computational cost prohibitive, even for moderately sized problems. To avoid the computational burden, a two-step conjugate gradient fast Fourier transform (CG-FFT) procedure is proposed. Such an approach reduces the computational cost and storage requirements of the CGM implementation. The efficient CGM is found to share a computational complexity similar to the distorted-Born iterative method (DBIM). Thus the convergence speed and accuracy of the CGM is compared with the DBIM. Numerical tests using both synthetic and experimental data show effectiveness for solving 2D inverse scattering problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a problem inspired by the real‐world need to identify the topographical features of ocean basins. Specifically, we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free‐surface deviation alone, within the context of dispersive shallow‐water wave models. The need to consider the shallow‐water regime arises from the ill‐posed nature of the problem and is motivated by prior work. Assuming only a relatively inaccurate initial guess for the bottom‐boundary, we design an algorithm to accurately deduce the fluid velocities and the true bottom‐boundary profile. We achieve this by considering two separate inverse problems: one to deduce the bottom‐boundary from velocities and the surface deviation, and another to recover the velocities from the surface deviation and an approximate bottom‐boundary. The former is a classic inverse problem that requires the inversion of an ill‐conditioned matrix, whereas the latter employs the observer framework. Combining the two inverse problems leads to our reconstruction algorithm. We emphasize the role played by model selection and its impact on algorithm design and the accuracy of the reconstruction. [ABSTRACT FROM AUTHOR]

Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another. [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, we further Meirong Zhang et al.'s work by computing the number of weighted eigenvalues for Sturm–Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm–Liouville equation with a general integrable potential, if its weight is a positive linear combination of n Dirac delta functions, then it has at most n (may be less than n, or even be 0) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly n. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm–Liouville problem with Dirac weights and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichlet problems for Sturm–Liouville equations involving single Dirac distribution weights is studied. [ABSTRACT FROM AUTHOR]

In an inverse combinatorial optimization problem, a feasible solution is given and is not optimal under the current parameters. The aim is to make the feasible solution optimal by modifying the current parameters as little as possible. The modification cost can be measured by different norms, such as weighted l 1 norm, weighted l 2 norm, weighted l ∞ norm, weighted Hamming distance and so on. In this paper, we focus on the constrained inverse minimum flow problems under the weighted Hamming distance. Three different models are considered: the general problem under the weighted bottleneck-type Hamming distance and two cases under the mixed of sum-type and bottleneck type. Strongly polynomial algorithms are presented for the models we studied. [ABSTRACT FROM AUTHOR]

Least squares form one of the most prominent classes of optimization problems with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty of the problem and motivates the need for efficient algorithms amenable to large-scale implementations. In this paper, we propose and analyze a Levenberg--Marquardt algorithm for nonlinear least squares subject to nonlinear equality constraints. Our algorithm is based on inexact solves of linear least-squares problems that only require Jacobian-vector products. Global convergence is guaranteed by the combination of a composite step approach and a nonmonotone step acceptance rule. We illustrate the performance of our method on several test cases from data assimilation and inverse problems; our algorithm is able to reach the vicinity of a solution from an arbitrary starting point and can outperform the most natural alternatives for these classes of problems. [ABSTRACT FROM AUTHOR]

This paper investigates the estimate of motion parameters from 3D hand joint positions. We formulate the issue as an inverse kinematics problem with biomechanical constraints and propose a fast and robust iterative approach to address the constrained optimization. It elaborately designs a coordinate descent algorithm to decompose the problem into a sequence of decisions on the transformation around each kinematic node (i.e., joint), while the decision for each node is equivalent to a point matching problem. Addressing the whole optimization then amounts to considering all nodes of the kinematic tree from its root to leaves one by one. This not only accelerates the process but also improves the accuracy of the solution of the inverse kinematic optimization. Experiments show that our approach is able to yield results comparable to and even better than those by the state-of-the-art 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]

The paper presents a comparative analysis of three fundamentally different algorithms for solving inverse problems of monitoring the layer thicknesses of optical coatings based on the data of monochromatic measurements of the reflection/transmission coefficients during deposition process. The previously developed geometric approach to the study of the thickness error correlation of deposited coatings is extended to the case of monochromatic measurements. A new parameter called the self-compensation factor was introduced to estimate the effect of error self-compensation. Its role in assessing the prospects for using various algorithms for coating deposition monitoring is shown. [ABSTRACT FROM AUTHOR]

The split feasibility problem is to find a point x∗ with the property that x∗∈ C and Ax∗∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem. [ABSTRACT FROM AUTHOR]

A new efficient random search algorithm is introduced for solving inversion problems in geophysical studies. The proposed algorithm is inherently a stochastic optimization method which is built on the concept of gradient descending and Lévy flights. Therefore, the algorithm is referred to as the Lévy gradient descent (L-GD). In which, the Lévy flights is a special class of random walk which consists of many short steps along with a few large steps. Such movements are observed in a varying range of fields, including animals' foraging patterns, fluid dynamics, transport of light and so on. Meanwhile, the Lévy flights typically shows much higher speed in searching for sparsely located targets compared to the well-known Brownian walks, which make them preferable to drive random search algorithms. As shown in the paper, besides optimal solutions of the inverse problems, the L-GD algorithm could also produce estimations on the error distributions of the resultant model parameters. Following a detailed introduction of the methodology, parameter settings of the algorithm are discussed in length through statistical experiments. Subsequently, the proposed algorithm is evaluated using numeric tests and shows attracting properties of the global convergence and significant higher searching efficiency compared to commonly adopted stochastic optimization techniques in geophysical inversions. Moreover, the L-GD algorithm is applied to the inversions of gravity and seismic travel time data and has achieved the same accuracy as gradient-based optimization methods. Meanwhile, though error estimations generated by L-GD algorithm are essentially qualitative, they could still provide valuable information to help evaluating the resultant model parameters, which is of great importance for practical geophysical inversions. [ABSTRACT FROM AUTHOR]

Super-resolution (SR) algorithms have now become a bottleneck for several remote sensing applications. SR is a technique that enhances minute details of the image by increasing spatial resolution of imaging systems. SR overcomes the problems of conventional resolution enhancement techniques such as introduction of noise, spectral distortion, and lack of clarity in the details of the image. In this paper, a survey has been conducted since the inception of SR algorithm till the latest state-of-the-art SR techniques to elucidate the importance of the SR algorithms that lead to paradigm shifts in the last two decades revolutionizing toward visually pleasing high-resolution image. Inspired from the natural images, the algorithms addressing the SR problems such as ill-posed, prior and regularization problem, inverse problem, multi-frame problem and illumination and shadow problem in remote sensing applications are analyzed. For an intuitive understanding of the paradigm shifts, publicly available images are tested with representative paradigm shift SR algorithms. The result of this paradigm shift analysis is done both qualitatively and quantitatively in terms of blurs in the image, pattern clarity, edge strength, and super-resolving capability. The convergence of the natural image to the remote sensed image is critically analyzed. The challenges with possible solutions for super-resolving the remote sensed image are recommended. On experimentation, it is found that deep learning-based SR algorithms produces visually pleasing images retaining sharp edges, enhanced spatial data, and clarity in feature representation while zooming at a certain level beyond interest. [ABSTRACT FROM AUTHOR]

Electrical Impedance Tomography or EIT is an imaging technique that reconstructs the conductivity distribution in the interior of an object using electric currents. In this paper, we study the continuum model for EIT in a domain where the geometric inclusions are fixed and only the conductivity values inside these inclusions are unknown. We show analytically and numerically how the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, a quasi-Newton method, can be effective in solving this inverse conductivity problem for EIT. [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]

Video snapshot compressive imaging (SCI) is a new compressive imaging system that aims to recover multiple video frames from a single measurement. Recent plug-and-play (PnP) imaging methods have been used for solving SCI inverse problems by leveraging pre-trained deep Gaussian denoisers. Under the condition of diminishing noise levels, a necessary assumption for the fixed-point convergence of PnP imaging algorithms is that the plugged denoisers are bounded denoisers. However, it is difficult to prove that existing deep Gaussian denoisers meet this assumption due to the complex network architectures, limiting the convergence analysis of these algorithms. This paper aims to elaborate a bounded deep video denoiser to remedy such a gap. Concretely, we propose a denoiser using double tight frames dubbed as VDTF and plug it into the PnP framework to construct the PnP-VDTF algorithm. In VDTF, a constant network (CNet) is designed, wherein a Swin Transformer-based module and a 3D convolution module extract the spatial information of the global features and the temporal information of the input video, respectively. Theoretically, we provide a strict boundary proof of VDTF, and show that PnP-VDTF can generate fixed-point convergent trajectories. Experiments show that PnP-VDTF can achieve higher-quality reconstructions compared with benchmark SCI algorithms. • We propose a provable and trainable bounded denoiser dubbed as VDTF. • Spatial–temporal information is utilized to generate adaptive thresholds. • The proposed PnP-VDTF algorithm can achieve high-quality reconstructions. • We explicitly prove that VDTF is a bounded denoiser. • We prove that PnP-VDTF can generate fixed-point convergent trajectories. [ABSTRACT FROM AUTHOR]

This paper provides a theoretical and numerical comparison of classical first-order splitting methods for solving smooth convex optimization problems and cocoercive equations. From a theoretical point of view, we compare convergence rates of gradient descent, forward-backward, Peaceman-Rachford, and Douglas-Rachford algorithms for minimizing the sum of two smooth convex functions when one of them is strongly convex. A similar comparison is given in the more general cocoercive setting under the presence of strong monotonicity and we observe that the convergence rates in optimization are strictly better than the corresponding rates for cocoercive equations for some algorithms. We obtain improved rates with respect to the literature in several instances by exploiting the structure of our problems. Moreover, we indicate which algorithm has the smallest convergence rate depending on strong convexity and cocoercive parameters. From a numerical point of view, we verify our theoretical results by implementing and comparing previous algorithms in well established signal and image inverse problems involving sparsity. We replace the widely used ℓ 1 norm with the Huber loss and we observe that fully proximal-based strategies have numerical and theoretical advantages with respect to methods using gradient steps. [ABSTRACT FROM AUTHOR]

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]

This paper investigates the potential of the concomitant use of both Isogeometric Boundary Element Method (IGABEM) and Artificial Neural Networks Algorithm (ANN) to determine the number, position and geometric shapes of voids in a plate subjected to lateral pressure. In the proposed approach the boundary conditions are given, and the displacements of a finite number of points provide the information required to define the geometric characteristics of one or more internal voids. Exploiting the potentialities of IGABEM, it is possible to achieve also complex geometries with a level of accuracy unthinkable with the shape functions commonly used in other numerical methods. Besides, the richness of the space of configurations obtainable with the isogeometric approach can be successfully handled by the ability of ANN to solve inverse problems with a high level of complexities. The concurrent use of both returns a powerful tool whose potentialities in solving inverse problems are here explored and discussed. [Display omitted] • An identification process is developed to identify internal voids in a 2D finite body. • Potential of Isogeometric BEM and ANN is investigate in identification process. • Tolerance of ANN for uncertainty and approximation allows use in complex problems. [ABSTRACT FROM AUTHOR]

In this paper, we systematically investigate the nonlocal Hirota equation with nonzero boundary conditions via Riemann–Hilbert method and multi-layer physics-informed neural networks algorithm. Starting from the Lax pair of nonzero nonlocal Hirota equation, we first give out the Jost function, scattering matrix, their symmetry and asymptotic behavior. Then, the Riemann–Hilbert problem with nonzero boundary conditions are constructed and the precise formulae of N -double poles solutions and N -simple poles solutions are written by determinants. Different from the local Hirota equation, the symmetry of scattering data for nonlocal Hirota equation is completely different, which results in disparate discrete spectral distribution. In particular, it could be more complicated and difficult to obtain the symmetry of scattering data under the circumstance of double poles. Besides, we also analyze the asymptotic state of one-double poles solution as t → ∞. Whereafter, the multi-layer physics-informed neural networks algorithm is applied to research the data-driven soliton solutions of the nonzero nonlocal Hirota equation by using the training data obtained from the Riemann–Hilbert method. Most strikingly, the integrable nonlocal equation is firstly solved via multi-layer physics-informed neural networks algorithm. As we all know, the nonlocal equations contain the PT symmetry P : x → − x , or T : t → − t , which are different with local ones. Adding the nonlocal term into the neural network, we can successfully solve the integrable nonlocal Hirota equation by multi-layer physics-informed neural networks algorithm. The numerical results show that the algorithm can recover the data-driven soliton solutions of the integrable nonlocal equation well. Noteworthily, the inverse problems of the integrable nonlocal equation are discussed for the first time through applying the physics-informed neural networks algorithm to discover the parameters of the equation in terms of its soliton solution. • We study the nonlocal Hirota equation with NZBCs, which is more complicated than one with ZBCs. • We give a more complex case of double poles for the nonlocal Hirota equation under ZBCs. • We study this nonlocal system with the PINN method for the first time. [ABSTRACT FROM AUTHOR]

This paper is devoted to recovering a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation with the time Caputo derivative from an additional integral condition. The uniqueness and a conditional stability for such an inverse problem are proved. Then the two-point gradient method is used to solve the inverse zeroth-order coefficient problem numerically. Some properties of the forward operator are obtained, such as the Fréchet differentiability, the Lipschitz continuity and the tangential cone condition to guarantee the convergence of the proposed algorithm. Four numerical examples in one-dimensional and two-dimensional spaces are provided to show the effectiveness and stability of the suggested algorithm. [ABSTRACT FROM AUTHOR]

• We propose an adaptive variational model based on saturation and con- trast enhancement for single image dehazing. • An adaptive function is introduced to balance the enhancement between saturation and contrast to prevent excessive artifacts. • A split Bregman algorithm is employed to accomplish an efficient numer- ical implementation for the proposed variational model. • Experimental comparisons demonstrate that the proposed method is com- parable to some popular dehazing methods and even outperforms some of them both qualitatively and quantitatively. Haze removal, also known as image dehazing, plays an important role in image/video processing and computer vision applications. Its main purpose is to eliminate haze to improve the visibility and quality of the image. However, due to the lack of scene information, including depth, transmission, and atmospheric light, haze removal essentially becomes a very challenging inverse problem. In this paper, we propose a novel and highly effective variational model for single image haze removal. The proposed method mainly utilizes a new idea of joint contrast and saturation enhancement. Incorporated with both adaptivity and physical cues, the proposed model is able to produce clear and visually satisfactory dehazed results and simultaneously prevent excessive artifacts caused by over-enhancement. The split Bregman algorithm is employed to accomplish an efficient numerical implementation for the proposed variational model. A number of numerical experiments and comparisons with some representative methods are conducted to demonstrate the good performance of the newly proposed model. [ABSTRACT FROM AUTHOR]

• Develop an effective and efficient ADMM solver for sparse phase retrieval with convergence guarantee. • Though the problem is nonconvex, the solver numerically solves it without requirement of good initialization. • The solver outperforms existing solvers in terms of the required measurement complexity. Phase retrieval, as a representative nonlinear inverse problem, is of increasing interest recently for its broad applications in imaging and engineering. Prediction of the signal amounts to solving a nonconvex optimization problem, which is generally NP-hard to solve. Collecting more measurements or providing prior information of the unknown are two often-seen strategies to facilitate the problem. In this paper, we propose an efficient and effective algorithm to solve phase retrieval with certain prior of the signal, in particular the signal itself is sparse in the natural basis. By formulating phase retrieval (PR) problem in the splitting form, we propose ADMM (alternating direction method of multipliers) with convergence guarantee to tackle the resulting nonconvex problem. Even though the algorithm is not new and there also exist several works on the convergence of ADMM, we clarify that our formulation for phase retrieval is not a specific application example of their investigated models. Hence, we investigate the convergence of our algorithm and show that ADMM converges a stationary point when penalty parameter ρ is large enough. It is observed that the recovery performance degrades when the penalty parameter increases. For better performance, we propose a practical scheme of tuning the penalty parameter ρ. Demonstration of the superior recovery performance on sparse phase retrieval (SPR) is conducted and it shows that our method numerically infers near-exact solution without providing good initialization. Our proposed method distinguishes itself from other existing competitive algorithms in two aspects: (a) the initialization is much less crucial for the algorithmic success than other compared methods; (b) the sampling complexity for the phase transition of recovery is much markedly reduced than other existing methods. [ABSTRACT FROM AUTHOR]

• Finite-difference approximation of the Sturm-Liouville problem with frozen argument is studied. • Theory of inverse spectral problems for the finite-difference approximation is developed. • Uniqueness theorems are proved and reconstruction algorithms are obtained for the degenerate and non-degenerate cases. • Numerical algorithm for solving the Sturm-Liouville inverse problem with frozen argument is developed. • Effectiveness of the algorithm is illustrated by numerical examples. This paper deals with the discrete system being the finite-difference approximation of the Sturm–Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the "correction terms" for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm–Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

• The adaptive L 2 norm is introduced to constrain the coefficient distribution of infrared image. • The alternation minimization algorithm is exploited to optimize the proposed BRWTR model. • The proposed model can well remove the random spectral noise, and produce the sharp and clean edge details. This paper proposes a novel image restoration model with wavelet-transform-based regularization for recovering high-resolution infrared images in students' training in the wild. The degraded infrared images are often contaminated by random Gaussian noises. The proposed method integrates the wavelet-transform-based prior, togethers with the adaptive L 2 norm constraint into the image reconstruction framework. The major novelty of this study is that wavelet-transform regularization is proposed for infrared blindly deconvolving images, which can save the image structures effectively. The famous alternation minimization algorithm is introduced to estimate the latent infrared images and instrument function. Experimental results demonstrate that the proposed BRWTR method can remove the random noise as well as produce the smooth results simultaneously. The qualitative comparison on the real infrared images with the state-of-the-art methods show the good performance of the proposed method. [ABSTRACT FROM AUTHOR]

Increasing the resolution of an image is an actual and extensively studied problem in image processing. Recently, Regularization by Denoising (RED) showing that any inverse problem can be handled by sequentially applying image denoising steps, including the image super-resolution (SR) task, which facilitate the resolution of the encountered optimization problem. In this paper, we propose a new configuration of genetic algorithms to resolve the super-resolution problem using a Non-Local Means filter as a denoiser function with a rigorous proof of the existence of a unique minimizer. In fact, since the SR algorithms always skip the complex spatial interactions within images, a more consistent model is then needed. The use of the genetic algorithms with the RED techniques guaranteed, in high intensity of noise and blur, the convergence to the globally optimal solution. As a result, the proposed algorithm shows efficient and consistent results, in terms of edges and feature preservation, compared with other SR approaches. • We treat the multi-frame super-resolution task based on genetic algorithms. • We propose the use of Nonlocal Means in the regularization term of the Regularization by Denoising cost function. • The use of genetic algorithms as an optimization approach to search the minimum of the proposed cost function gives better SR reconstruction result compared to other SR techniques. • The proposed method achieves superior computational performance in terms of PSNR and SSIM measures compared with other approaches. [ABSTRACT FROM AUTHOR]

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given. [ABSTRACT FROM AUTHOR]

The size-dependent properties of magnetic nanoparticles (MNP) are the major characteristics, determining MNP application in modern technologies and bio-medical techniques. Direct measurements of the nanosized particles, involved in intensive Brownian motion, are very complicated; so the correct mathematical methods for the experimental data processing enable to successfully predict the properties of MNP suspensions. In the present paper, we describe the fast numerical algorithm allowing to get the distribution over the relaxation time of MNP magnetic moments in ferrofluids. The algorithm is based on numerical fitting of the experimentally measured frequency spectra of the initial dynamic magnetic susceptibility. The efficiency of the algorithm in the solution of the inverse problem of magnetic granulometry is substantiated by the computer experiments for mono- and bi-fractional ferrofluids. [ABSTRACT FROM AUTHOR]

Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, the associated forward problem describes the evolution of the waves in the presence of known objects. This gives a symmetry relation: if the true objects (the desired solution of the inverse problem) were considered for solving the forward problem, then the recorded data should be returned. In this paper, we develop a topological derivative-based multifrequency iterative algorithm to reconstruct objects buried in attenuating media with limited aperture data. We demonstrate the method on half-space configurations, which can be related to problems set in the whole space by symmetry. One-step implementations of the algorithm provide initial approximations, which are improved in a few iterations. We can locate object components of sizes smaller than the main components, or buried deeper inside. However, attenuation effects hinder object detection depending on the size and depth for fixed ranges of frequencies. [ABSTRACT FROM AUTHOR]