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This paper is a continuation of the author’s previous papers [Front. Math. China, 2016, 11(6): 1379-1418; 2017, 12(5): 1023-1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context (the p-Laplacian operators for instance). Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case (p = 2) is obtained, which improves the author’s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities. [ABSTRACT FROM AUTHOR]

To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with linear transformation to the normalized unknowns and by the error propagation law to their variance-covariance matrix. A direct method (Direct Least Squares, DLS) is used to compute the initial values of LLS for reducing iteration number (to 1-3 cycles) with a 1.E-7 threshold in the paper. And, when the solution of 3D-ST having large rotational angles and uncertainties, it is also shown that DLS goes to a worse solution than LLS by means of a simulated transformation problem. [ABSTRACT FROM AUTHOR]

This paper deals with adaptively preconditioned iterative methods for solving large and sparse systems of linear equations. In particular, the paper discusses preconditioning where adaptive dropping reflects the quality of preserving the relation U Z = I between the direct factor U and the inverse factor Z that satisfy A = U T U and A − 1 = Z Z T . The proposed strategy significantly extends and refines the approach from [1], see also [2], by using a specific multilevel framework. Numerical experiments with two levels demonstrate that the new preconditioning strategy is very promising. Namely, we show a surprising fact that in our approach the Schur complement is better to form in a more sophisticated way than by a standard sparse matrix-matrix multiplication. [ABSTRACT FROM AUTHOR]

In this paper, we study a special instance of the split inverse problem (SIP), which is the split variational inclusion problem (SVIP). Three simple iterative methods for solving it are introduced and weak and strong convergence theorems are established under mild and standard assumptions. As an application, the problem of minimizing two proper, convex, and lower semi-continuous functions is considered. We compare and illustrate the efficiency and applicability of our schemes for several numerical experiments as well as an example in the field of compressed sensing. [ABSTRACT FROM AUTHOR]

In this paper, we will study the inverse problem of identification of flexural rigidity coefficient in the Euler-Bernoulli equation. This inverse problem is ill-posed. To solve it, we will use regularization methods. In particular, we will apply the mollification method and the Landweber iteration method, in particular, to find the regularized solution of the Moore- Penrose generalized inverse to a linear operator and with this, we reconstruct the coefficient. At the end of this paper, will present some examples of interest. [ABSTRACT FROM AUTHOR]

In this paper we derive convergence rates results for Landweber iteration in Hilbert scales. The assumptions that are necessary to prove these results are less restrictive than the ones given in an earlier paper. The relaxed conditions enlarge the range of applicability to a much wider class of nonlinear problems. The theory is applied to nonlinear Hammerstein operators. [ABSTRACT FROM AUTHOR]

Abstract The patient-specific biomechanical analysis of the aorta requires the quantification of the in vivo mechanical properties of individual patients. Current inverse approaches have attempted to estimate the nonlinear, anisotropic material parameters from in vivo image data using certain optimization schemes. However, since such inverse methods are dependent on iterative nonlinear optimization, these methods are highly computation-intensive. A potential paradigm-changing solution to the bottleneck associated with patient-specific computational modeling is to incorporate machine learning (ML) algorithms to expedite the procedure of in vivo material parameter identification. In this paper, we developed an ML-based approach to estimate the material parameters from three-dimensional aorta geometries obtained at two different blood pressure (i.e., systolic and diastolic) levels. The nonlinear relationship between the two loaded shapes and the constitutive parameters is established by an ML-model, which was trained and tested using finite element (FE) simulation datasets. Cross-validations were used to adjust the ML-model structure on a training/validation dataset. The accuracy of the ML-model was examined using a testing dataset. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a multilevel method for solving nonlinear inverse problems F (x) = y in Banach spaces. By minimizing the discretized version of the regularized functionals at different levels, we define a sequence of regularized approximations to the sought solution, which is shown to be stable and globally convergent. The penalty term Θ in regularized functionals is allowed to be non-smooth to include L p − L 1 or L p − TV (Total Variation) reconstructions, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. Two parameter identification examples are presented to validate the theoretical analysis and to verify the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, yk=ak∗x for k=1,…,m, is it possible to recover x∈Cn (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors ak’s are generic (i.i.d. complex Gaussian) and numerous enough (m≥Cnlog3n), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal x, up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem. It is proved that the proposed algorithm always converge to the generalized Hamiltonian solutions with a submatrix constraint of Problem 1.1 within finite iterative steps in the absence of roundoff error. In addition, by choosing a special kind of initial matrices, it is shown that the minimum norm solution of Problem 1.1 can be obtained consequently. At last, for a given matrix group in the solution set of Problem 1.1 , it is proved that the unique optimal approximation solution of Problem 1.2 can be also obtained. Some numerical results are reported to demonstrate the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

The non-negative regularization problem has been widely studied for finding non-negative sparse solutions of linear inverse problems and gained successful applications in various application areas. In the present paper, we propose an iterative positive thresholding algorithm (IPTA) to solve the non-negative regularization problem and investigate its convergence properties in finite- or infinite-dimensional Hilbert spaces. The significant advantage of the IPTA is that it is very simple and of low computation cost, and thus, it is practically attractive, especially for large-scale problems. The global convergence of the IPTA is achieved under some mild assumptions on algorithmic parameters. Furthermore, we introduce a notion of positive orthogonal sparsity pattern, and use it to establish the linear convergence rate of the IPTA to a global minimum. Finally, the numerical study on compressive sensing shows that the proposed IPTA is efficient in approaching the non-negative sparse solutions of linear inverse problems and outperforms several existing algorithms in sparse optimization. [ABSTRACT FROM AUTHOR]

In this paper, a new computer-time effective iterative method is proposed to enhance the spatial resolution of microwave radiometer data in both Hilbert and Banach spaces. The reconstruction method is based on the conjugate gradient (CG) method. CG is a well-known method in electromagnetics, where it is commonly used to address inverse problems in Hilbert spaces. However, the method has never been applied to inverse problems related to spatial resolution enhancement of microwave remotely sensed measurements. In this paper, CG is adapted to the microwave radiometer case and exploited to reconstruct the brightness field at enhanced spatial resolution. Then, CG is, for the first time, extended to Banach spaces by means of duality maps and then applied to enhance the radiometer spatial resolution. Experiments undertaken on both simulated and actual radiometer data confirm the soundness of the proposed approach in Banach spaces and demonstrate that CG is able to provide a reconstruction accuracy similar to the conventional Landweber method, but with a significantly reduced processing time. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the numerical identification of the fractional parameters in the fractional mobile/immobile advection-diffusion model. For the direct problem, two efficient numerical schemes with second-order spatial accuracy and fourth-order spatial accuracy are derived, respectively. The unconditional stability and convergence of the numerical schemes are analysed rigorously by the Fourier analysis method. For the inverse problem, we first propose a novel numerical scheme to compute the fractional sensitivity matrix, and then employ the nonlinear Levenberg-Marquardt iterative method to estimate the fractional parameters. Finally, numerical examples are given to illustrate the effectiveness and accuracy of the proposed numerical schemes, and the numerical identification of the fractional parameters is also presented in tabular form, which demonstrate the effectiveness of the proposed numerical method for the identification of the fractional parameters in the fractional mobile/immobile advection-diffusion model. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements. [ABSTRACT FROM AUTHOR]

The inverse coefficient problems for estimating the unknown space-time dependent conductivity function and reaction coefficient function of a time-fractional diffusion reaction equation for non-uniform material are solved in the paper, without needing of initial condition, final time condition and internal measurement. We tackle these inverse coefficient problems supplementing with boundary data. After a homogenization technique by using these boundary data, a sequence of boundary functions are derived, which together with the zero element constitute a linear space. As an approximation, a boundary functional is proved in the linear space, of which the time-dependent energy is preserved for the energetic boundary function at each time step. The linear systems and iterative algorithms used to recover the unknown conductivity function and reaction coefficient with energetic boundary functions as bases are developed, which are convergent fast at each time step. The data required are merely the boundary temperatures and fluxes, and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the estimated results under large noise with exact solutions. [ABSTRACT FROM AUTHOR]

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are “exact” in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system. [ABSTRACT FROM AUTHOR]

In this paper we consider discrete inverse problems for which noise becomes negligible compared to data with increasing model norm. We introduce two novel definitions of regularization for characterizing inversion methods which provide approximations of ill-conditioned inverse operators consistent with such noisy data. In particular, these d efinitions, respectively, require that the reconstruction error computed from normalized data (p-asymptotic regularization) and the relative reconstruction error (p-relative regularization) go to zero as the model norm tends to infinity, 0 ≤ p < 1 being a parameter controlling the increase rate of the noise level. We investigate the relationship between these two definitions and we prove that t hey are all equivalent for positively homogeneous iterative algorithms with suitable stopping rules. This result has as a crucial consequence that such iterative algorithms realize regularization independently of the noise model. Then we give sufficient conditions for such methods to be p-asymptotic and p-relative regularizations in a discrete setting and we prove that the classical expectation maximization algorithm for Poisson data and the Landweber algorithm, if suitably stopped, are regu larization methods in this sense. We perform numerical simulations in the case of image deconvolution and computerized tomography to show that, in the presence of model-dependent noise, the reconst ructions provided by the above mentioned methods improve with increasing model norm as required by the p-asymptotic and p-relative regularization properties. More extensive studies on the p-asym ptotic and p-relative regularizations for Tikhonov-type methods will be the object of future work. [ABSTRACT FROM AUTHOR]

In this paper, we first provide semi-convergence analysis for a special GPIU(Generalized Parameterized Inexact Uzawa) method with singular preconditioners for solving singular saddle point problems. We next provide a methodology of how to choose nearly quasi-optimal parameters of the special GPIU method. Lastly, numerical experiments are carried out to examine the effectiveness of the special GPIU method with singular preconditioners by comparing its performance with that of other existing iterative methods for solving singular saddle point problems. [ABSTRACT FROM AUTHOR]

In this paper we consider the "exterior approach" to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification. [ABSTRACT FROM AUTHOR]

Localizing sources in physical systems represents a class of inverse problems with broad scientific and engineering applications. This paper is concerned with the development of a non-iterative source sensitivity approach for the localization of sources in linear systems under steady-state. We show that our proposed approach can be applied to a broad class of physical problems, ranging from source localization in elastodynamics and acoustics to source detection in heat/mass transport problems. The source sensitivity field introduced in this paper represents the change of a cost functional caused by the appearance of an infinitesimal (or point) source in a given domain (or its boundary). In order to extract macroscopic inferences, we apply a threshold to the source sensitivity field in a way that parallels the application of the topological derivative concept in shape identification. We establish precise formulas for the source sensitivity field using a direct approach and a Lagrangian formulation. We show that computing the source sensitivity field entails just obtaining the solution of a single adjoint problem. Hence, the computational expense of obtaining the source sensitivity is of the same order as that of solving one forward problem. We illustrate the performance of the method through numerical examples drawn from the areas of elastodynamics, acoustics, and heat/mass transport. Our results show that our proposed approach could be used on its own as a source detection tool or to obtain initial guesses for more quantitative iterative gradient-based minimization strategies. [ABSTRACT FROM AUTHOR]

Electrical capacitance tomography (ECT) is considered a promising visualization measurement technique, in which image reconstruction algorithms play an important role in real applications. In this paper, a dynamic reconstruction model, which integrates the ECT measurement information and the dynamic evolution information of the reconstruction objects, is presented. A generalized objective function, which simultaneously considers the ECT measurement information, the dynamic evolution information of the objects of interest, the temporal constraints and the spatial constraints, is proposed. An iteration scheme that integrates the beneficial advantages of the split Bregman iteration technique and the homotopy algorithm is developed for solving the proposed objective function. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. For the cases simulated in this paper, the accuracy of the images reconstructed by the proposed algorithm is improved and the artifacts in the reconstructed images can be removed effectively, which indicates that the proposed algorithm is successful in solving ECT inverse problems. [ABSTRACT FROM AUTHOR]

Abstract: Owing to distinct advantages such as the easy implementation, low cost, high safety and non-intrusive sensing, the electrical capacitance tomography (ECT) is considered as a promising visualization measurement technique, in which reconstructing high quality images is highly desirable for real applications. In this paper, a multi-scale dynamic reconstruction model, which simultaneously utilizes the ECT measurement information and the dynamic evolution information of a dynamic object, is presented. The original dynamic image reconstruction problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the original scale. A generalized objective functional that considers the ECT measurement information, the dynamic evolution information of a dynamic object, the temporal constraint and the spatial constraint is proposed. An iterative algorithm, which integrates the beneficial advantages of the evolutionary strategy (ES) algorithm and the homotopy method, is designed for solving the proposed objective functional. Numerical simulations are implemented to evaluate the feasibility of the proposed algorithm. For the cases simulated in this paper, the quality of the images reconstructed by the proposed algorithm is improved, which indicates that the proposed algorithm is successful in solving ECT inverse problems. [Copyright &y& Elsevier]

One of the main computational drawbacks in the application of 3-D iterative inversion techniques is the requirement of solving the field quantities for the updated contrast in every iteration. In this paper, the 3-D electromagnetic inverse scattering problem is put into a discretized finite-difference frequency-domain scheme and linearized into a cascade of two linear functionals. To deal with the nonuniqueness effectively, the joint structure of the contrast sources is exploited using a sum-of- \ell 1 -norm optimization scheme. A cross-validation technique is used to check whether the optimization process is accurate enough. The total fields are, then, calculated and used to reconstruct the contrast by minimizing a cost functional defined as the sum of the data error and the state error. In this procedure, the total fields in the inversion domain are computed only once, while the quality and the accuracy of the obtained reconstructions are maintained. The novel method is applied to ground-penetrating radar imaging and through-the-wall imaging, in which the validity and the efficiency of the method are demonstrated. [ABSTRACT FROM PUBLISHER]

This paper describes a new algorithm that solves a particular phase retrieval problem, with important applications in audio processing: the reconstruction of a function from its scalogram, that is, from the modulus of its wavelet transform. It is a multiscale iterative algorithm that reconstructs the signal from low-to-high frequencies. It relies on a new reformulation of the phase retrieval problem that involves the holomorphic extension of the wavelet transform. This reformulation allows to propagate phase information from low-to-high frequencies. Numerical results, on audio and non-audio signals, show that reconstruction is precise and stable to noise. The complexity of the algorithm is linear in the size of the signal, up to logarithmic factors. It can thus be applied to large signals. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear iterative methods, which are different from the classical algorithms, to solve inverse problems are presented. Our methods by denoting some parameters and some properties of the algorithm in both noise and noiseless cases are studied. Finally, the convergence of the sequence generated by the algorithm without noise is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we improved the performance of the contrast source inversion (CSI) method by incorporating a so-called cross-correlated cost functional, which interrelates the state error and the data error in the measurement domain. The proposed method is referred to as the cross-correlated CSI. It enables better robustness and higher inversion accuracy than both the classical CSI and multiplicative regularized CSI (MR-CSI). In addition, we show how the gradient of the modified cost functional can be calculated without significantly increasing the computational burden. The advantages of the proposed algorithms are demonstrated using a 2-D benchmark problem excited by a transverse magnetic wave as well as a transverse electric wave, respectively, in comparison with classical CSI and MR-CSI. [ABSTRACT FROM PUBLISHER]

In this paper, a non-iterative technique is proposed for the transverse load identification on Kirchhoff plates using approximate Green’s functions (AGFs). In this way, we firstly employ the recently introduced meshless method to construct the AGFs, as the combination of a series of Trefftz basis, i.e. Exponential basis functions (EBFs), and the fundamental solutions of the governing equation. As will be explained, using a proper set of EBFs, as well as a collocation technique, enables us to construct the AGFs for different types of domain shape and boundary conditions. In the second step, a set of artificially generated results, in the absence of realistic experimental results, are used to express the plate’s response field, i.e. deflection or velocity fields, as a series of AGFs through a collocation technique. It will be shown how the constant coefficients of the response series are related to the intensity of the reconstructed force at a set of selected points. The proposed method is capable of constructing both distributed and concentrated loads with desirable accuracy. This ability is shown in the solution of three sample problems of the static and time-harmonic force recovery. [ABSTRACT FROM PUBLISHER]

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz []. We here present a convergence analysis of the overall method as well as numerical experiments. [ABSTRACT FROM AUTHOR]

Parameter identification problems typically consist of a model equation, e.g., a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularizat ion method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this paper we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type, and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-once and reduced versions and provide some numerical comparison. [ABSTRACT FROM AUTHOR]

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with a general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of a Lagrange multip lier. In case the data contains noise, we show that our method is a regularization method as long as it is terminated by a suitable stopping rule. Various numerical simulations are performed to test the efficiency of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the projection algorithm studied by Iiduka and Takahashi (2008) for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space. We first remark that, under the assumptions imposed on the operator in their paper, the iterative sequence converges weakly to a zero of the operator, not just a solution of the variational inequality problem. In our proof, slightly modified from the original, we do not assume the uniform smoothness of a space as was the case there. Finally, using Halpern’s type method, we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the Bergman distance associated with the function . [Copyright &y& Elsevier]

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]

The purpose of this paper is to introduce and study a modified Halpern's iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper improves and extends some recent results done by Xu (Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018) and some others. [ABSTRACT FROM AUTHOR]

Abstract: This paper aims to reduce the problems of incomplete data in computed tomography, which happens frequently in medical image process and analysis, e.g., when the high-density region of objects can only be penetrated by X-rays at a limited angular range. As the projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. Image reconstruction based on total variation (TV) reduces the problem and gives better performance on edge-preserving reconstruction; however, the artificial parameter can only be determined through considerable experimentation. In this paper, an effective TV objective function is proposed to reduce the inverse problem in the limited angle tomography. This novel objective function provides a robust and effective reconstruction without any artificial parameter in the iterative processes, using the TV as a multiplicative constraint. The results demonstrate that this reconstruction strategy outperforms some previous ones. [Copyright &y& Elsevier]

Abstract: Electrical capacitance tomography (ECT) is considered as a promising tomography technology, and exactly reconstructing the original objects is highly desirable in real applications. In this paper, a generalized image reconstruction model that simultaneously considers the inaccurate property in the measured capacitance data and the linearization approximation error is presented. A generalized objective function, which has been developed using a combinational M-estimation and an extended stabilizing item, is proposed. The objective function unifies six estimation methods into a concise formula, where different estimation methods can be easily obtained by selecting different parameters. The homotopy method that integrates the beneficial advantages of the alternant iteration scheme is employed to solve the proposed objective function. Numerical simulations are implemented to evaluate the numerical performances and effectiveness of the proposed algorithm, and the numerical results reveal that the proposed algorithm is efficient and overcomes the numerical instability in the process of ECT image reconstruction. For the reconstructed objects in this paper, a dramatic improvement in accuracy and spatial resolution can be achieved, which indicates that the proposed algorithm is a promising candidate for solving ECT inverse problems. [Copyright &y& Elsevier]

The optical tomography problem presents some interesting difficulties for both, experimental and theoretical work. This paper has attempted an overview of the theoretical problems for image reconstruction. In this paper we review some general approaches to inverse problems to set the context for optical tomography. An essential requirement is to treat the problem in a nonlinear fashion, by using an iterative method. The inverse problem is approached by numerical solutions methods using MathCad program. The Radon transform is the basic tool of the computerized tomography. In the sequel we introduce this transform, review some properties and present a numerical program for its inversion. We show some results that represent the most complex and realistic simulations of optical tomography yet developed. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a method to integrate basic origami processes into fractal concepts. The starting point is an area of polygonal shape. It is folded until one or more smaller copies result. The crease pattern is scaled to the reduced copies, and then the process repeats. This is equivalent to maintaining the original crease pattern if the reduced objects are magnified. As will become clear in the paper, the fact that the folded object is enlarged implies that functions are used that are the inverse of contracting functions, as is familiar in inverse techniques for iterated function systems. The method is generalized by allowing cuts before folding. With a well-chosen colorization method, the images obtained reveal complex structures suggestive of depth and light. [Copyright &y& Elsevier]

We give an explicit Krasnoselski-Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: {G(x*,y) ≥0, ∀x ϵFix(T), find x* ϵ Fix(T) such that ⟨X* - ƒ (x*), x - x*⟨ ≥ 0, ∀x ϵFix(T), where C is a closed convex subset of a Hilbert space H, G : C × C →ℝ is an equilibrium function, T : C → C is a nonexpansive mapping with Fix(T) its set of fixed points and ƒ : C → C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, 'Weak and strong convergence of Krasnosel'skiĭ-Mann iteration for hierarchical fixed point problems', Inverse Problems 24 (2008), 501-508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction ƒ was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra. [ABSTRACT FROM AUTHOR]

Superior to the other well-known iterations, such as Landweber iteration, total variational regularization is a well-established method for identifying the piecewise smooth parameter and processing the image. Therefore, we propose a new iteratively total variational regularization based on total variational regularization and homotopy perturbation technique in this paper. Taking advantage of Bregman distance, we construct a nested iteration to inverse parameters. Meanwhile we prove that the error of the new method decreases monotonically, and the new iterative method with the noisy data is regular according to the discrepancy principle. In the last numerical section, compared with Landweber type total variational regularization and Runge–Kutta type regularization, numerical results of this new method indicate that this new regularization is time saving for the same accuracy, and more robust to the noisy data. [ABSTRACT FROM AUTHOR]

Reconstruction of fields using partial measurements is of vital importance in different applications in geosciences. Solving such an ill-posed problem requires a well-chosen model. In recent years, training images (TI) are widely employed as strong prior models for solving these problems. However, in the absence of enough evidence it is difficult to find an adequate TI which is capable of describing the field behavior properly. In this paper a very simple and general model is introduced which is applicable to a fairly wide range of binary images without any modifications. The model is motivated by the fact that nearly all binary images are composed of simple linear edges in micro-scale. The analytic essence of this model allows us to formulate the template matching problem as a convex optimization problem having efficient and fast solutions. The model has the potential to incorporate the qualitative and quantitative information provided by geologists. The image reconstruction problem is also formulated as an optimization problem and solved using an iterative greedy approach. The proposed method is capable of recovering the image unknown values with accuracies about 90% given samples representing as few as 2% of the original image. [ABSTRACT FROM AUTHOR]

Patch near-field acoustical holography methods like statistically optimized near-field acoustical holography and equivalent source method are limited to relatively low frequencies, where the average array-element spacing is less than half of the acoustic wavelength, while beamforming provides useful resolution only at medium-to-high frequencies. With adequate array design, both methods can be used with the same array. But for holography to provide good low-frequency resolution, a small measurement distance is needed, whereas beamforming requires a larger distance to limit sidelobe issues. The wideband holography method of the present paper was developed to overcome that practical conflict. Only a single measurement is needed at a relatively short distance and a single result is obtained covering the full frequency range. The method uses the principles of compressed sensing: A sparse sound field representation is assumed with a chosen set of basis functions, a measurement is taken with an irregular array, and the inverse problem is solved with a method that enforces sparsity in the coefficient vector. Instead of using regularization based on the 1-norm of the coefficient vector, an iterative solution procedure is used that promotes sparsity. The iterative method is shown to provide very similar results in most cases and to be computationally much more efficient. [ABSTRACT FROM AUTHOR]

Tomographic iterative reconstruction methods need a very thorough modeling of data. This point becomes critical when the number of available projections is limited. At the core of this issue is the projector design, i.e., the numerical model relating the representation of the object of interest to the projections on the detector. Voxel driven and ray driven projection models are widely used for their short execution time in spite of their coarse approximations. Distance driven model has an improved accuracy but makes strong approximations to project voxel basis functions. Cubic voxel basis functions are anisotropic, accurately modeling their projection is, therefore, computationally expensive. Both smoother and more isotropic basis functions better represent the continuous functions and provide simpler projectors. These considerations have led to the development of spherically symmetric volume elements, called blobs. Set apart their isotropy, blobs are often considered too computationally expensive in practice. In this paper, we consider using separable B-splines as basis functions to represent the object, and we propose to approximate the projection of these basis functions by a 2D separable model. When the degree of the B-splines increases, their isotropy improves and projections can be computed regardless of their orientation. The degree and the sampling of the B-splines can be chosen according to a tradeoff between approximation quality and computational complexity. We quantitatively measure the good accuracy of our model and compare it with other projectors, such as the distance-driven and the model proposed by Long et al. From the numerical experiments, we demonstrate that our projector with an improved accuracy better preserves the quality of the reconstruction as the number of projections decreases. Our projector with cubic B-splines requires about twice as many operations as a model based on voxel basis functions. Higher accuracy projectors can be used to improve the resolution of the existing systems, or to reduce the number of projections required to reach a given resolution, potentially reducing the dose absorbed by the patient. [ABSTRACT FROM AUTHOR]

In this paper, a multi-step iterative method with seventh-order local convergence is presented, for solving nonlinear systems. A development of an inverse first-order divided difference operator for multivariable function is applied to prove the local convergence order of the new method. The computational efficiency is compared with some known methods. It is proved that the new method is more efficient. Numerical experiments are performed, which support the theoretical results. From the comparison with some known methods it is observed that the new method remarkably saves the computational time in the high-precision computing. [ABSTRACT FROM AUTHOR]

The image reconstruction of the electrical capacitance tomography (ECT) is an ill-posed and sparse problem. In order to increase the accuracy and speed of the image reconstruction, this paper proposes a new reconstruction algorithm which is based on the extreme learning machine (ELM) with the Landweber iteration method. Firstly, a nonlinear mapping model is established between the pixel gray-scale values and the interelectrode capacitances by using the ELM which has a good learning ability and high speed. Secondly, the Landweber iteration method, which has a good performance in convergence and stability, is applied to calculate the output weight matrix of ELM. Finally, a convergence and stable mapping model of ELM with the Landweber iteration algorithm (L-ELM) for ECT image reconstruction is trained on Matlab platform. Both simulation and measurement tests are carried out to evaluate and analyze the proposed method. Experimental results indicate that the proposed algorithm has good generalization ability and high image reconstruction quality which are better than those of conventional ELM algorithm. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is a promising technique that is capable of creating high-resolution subsurface images of the earth from seismic data. However, it is computationally expensive, especially when 3D elastic wave equation is considered. In this paper an adjoint-based computational algorithm has been proposed to address the computational challenges. The important strategy in this work is to decouple the two Lamé parameters so that two half-sized subproblems are resulted and solved separately. Mathematically, the inverse problem is formulated as a PDE-constrained optimization problem in which the objective functional is defined as the misfit between observational and synthetic data. The parameters to be recovered are the spatially varying Lamé parameters which are of great interests to geophysicists for the purpose of hydrocarbon exploration and subsurface imaging. The gradient of the misfit functional with respect to the Lamé parameters is calculated by solving the adjoint elastic wave equation, whilst the Quasi-Newton method (L-BFGS) is used to minimize the misfit functional. Numerical experiments demonstrated that the new method is accurate, efficient and robust in recovering Lamé parameters from seismic data. [ABSTRACT FROM AUTHOR]

Bregman iterative algorithms have been extensively used forand total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure. [ABSTRACT FROM AUTHOR]

The paper describes the method for solving the inverse problems of gravimetry based on the functional representations, which follow from the variational principles in the uniform metrics with respect to density models of the geological medium. The functional representations are obtained for both the linear problem (which study local density distributions) and nonlinear problem (which study a system of structural models). The explicit formulas for calculating density models are derived for the particular cases based on the introduced functional representations of the obtained solutions. In the general case, the converging iterative processes providing the solution for both the density distribution models and structural models are constructed. The relationship is established between the functional representations implementing the variational principle in the uniform metric and linear integral representations corresponding to the optimization in the quadratic norm, on one hand, and the other known density models, on the other hand. [ABSTRACT FROM AUTHOR]

In this paper, a method with very high order of convergence is constructed and analyzed. The method is used to compute generalized inverses. The efficiency index has been employed to show its superiority. Numerical experiments re-verify that the proposed iterative expression is more effective than the existing methods of the same type. [ABSTRACT FROM AUTHOR]

Many physics problems can only be studied by coupling various numerical codes, each modeling a subaspect of the physics problem that is addressed. In most cases, the “brute force” technique of running the codes one after the other in a loop until convergence is reached requires excessive CPU time. The present paper illustrates that re-writing the coupling as a root-finding problem, to which a quasi-Newton method–here the (Inverse) Column Updating Method–can be applied, is useful to push down the computation time, at the expense of a very modest amount of supplementary programming. A simplified version of the set of codes commonly used to describe plasma heating by radio frequency waves in a tokamak plasma is adopted for illustrating the potential of the speed-up method. It consists of a wave equation as well as a Fokker–Planck velocity space diffusion and a radial energy diffusion model. It is shown that with this approach a substantial reduction in CPU time needed for convergence can be obtained. [ABSTRACT FROM AUTHOR]