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In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known "fast iterative shrinkage-thresholding algorithm" (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless numerical procedure, the radial basis function-generated finite difference (RBF-FD), to approximate the TFCM involving two fractional temporal derivatives. The time discretization of the TFCM is performed based on the Grunwald–Letnikov expansion. The spatial derivatives are discretized using the RBF-FD. The pattern of data distribution in the support domain is assumed as having a fixed number of points. The RBF-FD is based on the local support domain that leads to a sparsity system and also tackles the ill-conditioning problem caused by global collocation method. The theoretical stability and the convergence analysis of the scheme are also discussed in detail. It is shown that the proposed method is efficient and that the numerical results confirm the theoretical formulation.

This paper considers different versions of block-column iterative (BCI) methods for solving nonnegative constrained linear least squares problems. We present the convergence analysis for a family of stationary BCI methods with nonnegativity constraints (BCI-NC), which is applicable to linear complementarity problems (LCP). We also consider the flagging idea for BCI methods, which allows saving computational work by skipping small updates. Also, we combine the BCI-NC algorithm and the flagging version of a nonstationary BCI method with nonnegativity constraints to derive a convergence analysis for the resulting method (BCI-NF). The performance of our algorithms is shown on ill-posed inverse problems taken from tomographic imaging. We compare the BCI-NF and BCI-NC algorithms with three recent algorithms: the inner-outer modulus method (Modulus-CG method), the modulus-based iterative method to Tikhonov regularization with nonnegativity constraint (Mod-TRN method), and nonnegative flexible CGLS (NN-FCGLS) method. Our algorithms are able to produce more stable results than the mentioned methods with competitive computational times.

The nonlinear modified anomalous sub-diffusion model characterizes processes that become less anomalous as time progresses by including a second fractional time derivative acting on the term of diffusion. This paper introduces a radial basis function-generated finite difference (RBF-FD) method for solving the governing problem. The Grunwald–Letnikov formula with first-order accuracy is implemented to discretize the problem in the time direction, and the spatial variable is discretized using the local RBF-FD method. The convergence and stability of the time discretization scheme are deduced in an appropriate Sobolev space. The data distribution pattern within the support domain is considered to have a constant number of points. The numerical results on regular and irregular domains show the efficiency and high accuracy of the method and confirm the theoretical prediction.

This paper presents an iterative procedure based on an implicit finite difference method to solve a mathematical model of drug delivery from a planar matrix with a moving boundary condition. This model includes the diffusion equation with space–time fractional-order derivatives. We establish the stability and convergence analysis of the method. We compare the numerical results with the scale-invariant and the homotopy perturbation solutions for different space–time-fractional orders and the problem parameters.

In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to $$ \aleph (u):\equiv u_{t}-\nabla \cdot (F(x,\nabla u)) $$ where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function F. At the first step of the methodology, we give a stability result corresponding to connectivity of F and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions.

The price variation of the correlated fractal transmission system is used to deduce the fractional Black–Scholes model that has an $$\alpha $$ -order time fractional derivative. The fractional Black–Scholes model is employed to price American or European call and put options on a stock paying on a non-dividend basis. Upon encountering fractional differential equations, the efficient and relatively reliable numerical schemes must be obtained for their solution due to fractional derivatives being non-local. The present paper is aimed at determining the numerical solution of the time fractional Black–Scholes model (TFBSM) with boundary conditions for a problem of European option pricing involved with the method of radial basis functions (RBFs), which is a truly meshfree scheme. The TFBSM is discretized in the temporal sense based on finite difference scheme of order $${\mathcal {O}}(\delta t^{2-\alpha })$$ for $$ 0< \alpha

This paper considers a wireless cooperative network with a base station (central unit) and many subscribers (users) in which the subscribers have the ability to relay information for each other to improve the overall network performance. In future wireless communication networks where subscribers are intellectual and pursue various objectives, they will not help relay information for others. This means that the viability of cooperative communication networks largely depends on the willingness of users to help. The reputation based schemes solved this problem by stimulating the users to cooperate with others. Besides, the average achievable rate at the destination node in a relay based network highly depends on the relay node location. Therefore, for a relay selection scheme, the choices of relay location, and the reputation of each user as well as signal-to-noise ratio, which is the main criterion for relay selection schemes, are important selection parameters. This paper proposes a novel multiple criteria relay selection scheme with a low implementation complexity. The proposed scheme incorporates reputation, SNR and location of the users in a highest-level design that not only selects the best relay node for each source-destination pair, but also assigns weight efficiently for each criterion. A simple and efficient optimization tool based on multiple criteria decision making method is employed to solve our formulated problem. In addition, a novel information theoretic measure, which effectively measures dissimilarity between candidate relay nodes and ideal solutions, is also investigated. Since the proposed scheme determines the potential relay nodes in order of their superiority based on the predetermined criteria, it can be utilized in multiple relay selection schemes in wireless relay networks and also in the indirect reciprocity game methods. Finally, the numerical results are provided to illustrate the comparisons between different typical experiments. To best of our knowledge, no prior work has studied the process of relay selection from this point of view.

A nonlinear wave phenomenon is one of the significant fields of scientific research, which a lot of researchers in the past have deliberated about mathematical models clarifying the treatment. The nonlinear Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers (KdV-BBM-B) model plays an essential role in numerous subjects of engineering and science. This model is numerically approximated by means of the combination of the radial basis function (RBF) and the finite difference (RBF-FD), RBF-pseudospectral (RBF-PS) that are stated in this paper. These methods are meshless, and no obligation is required to linearize the nonlinear parts. The radial kernels are employed to convert the PDE into a system of ODEs. This ODE system is solved with the help of the higher-order ODE solver. In addition, it is indicated that the present numerical techniques are stable. To evaluate the validity and applicability of the aforesaid techniques, the error norms and three invariants I1, I2 and I3 are calculated and displayed both numerically and graphically. The obtained results are compared with previous schemes found in the literature.

Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order $$\mathcal {O}(\delta t^{})$$ in the t-direction is described via the finite difference scheme for $$ 0< \alpha

Column-oriented versions of algebraic iterative methods are interesting alternatives to their row-version counterparts: they converge to a least squares solution, and they provide a basis for saving computational work by skipping small updates. In this paper we consider the case of noise-free data. We present a convergence analysis of the column algorithms, we discuss two techniques (loping and flagging) for reducing the work, and we establish some convergence results for methods that utilize these techniques. The performance of the algorithms is illustrated with numerical examples from computed tomography.

The fractal mobile/immobile model of the solute transport is based on the assumption that the waiting times in the immobile region follow a power-law, and this leads to the application of fractional time derivatives. The model covers a wide family of systems that include heat diffusion and ocean acoustic propagation. This paper develops an efficient computational technique, stemming from the radial basis function-generated finite difference (RBF-FD), to solve the fractal mobile-immobile transport model (FMTM). The time fractional derivative of the FMTM is discretized via the shifted Grunwald-Letnikov formula with second-order accuracy. On the other hand, the spatial derivative is approximated using the local RBF-FD method. The main benefit of the local collocation technique is that we only need to consider discretization points present in each of the sub-domains around the collocation point. The stability and convergence analysis of the proposed method are proven via the energy method in the L2 space. The numerical results for the FMTM on regular and irregular domains confirm the theoretical formulation and efficiency of the proposed scheme.

The elastography (elasticity imaging) is one of the recent state-of-the-art methods for diagnosis of abnormalities in soft tissue. The idea is based on the computation of the tissue elasticity distribution. This leads to the inverse elasticity problem; in that, displacement field and boundary conditions are known, and elasticity distribution of the tissue is aimed for computation. We treat this problem by the Gauss–Newton method. This iterative method results in an ill-posed problem, and therefore, regularization schemes are required to deal with this issue. The impacts of the initial guess for tissue elasticity distribution, contrast ratio between elastic modulus of tumor and normal tissue, and noise level of the input data on the estimated solutions are investigated via two different regularization methods. The numerical results show that the accuracy and speed of convergence vary when different regularization methods are applied. Also, the semi-convergence behavior has been observed and discussed. At the end, we signify the necessity of a clever initial guess and intelligent stopping criteria for the iterations. The main purpose here is to highlight some technical factors that have an influence on elasticity image quality and diagnostic accuracy, and we have tried our best to make this article accessible for a broad audience.

An algorithm for solving convex feasibility problem for a finite family of convex sets is considered. The acceleration scheme of De Pierro (em Methodos de projecao para a resolucao de sistemas gerais de equacoes algebricas lineares. Thesis (tese de Doutoramento), Instituto de Matematica da UFRJ, Cidade Universitaria, Rio de Janeiro, Brasil, 1981), which is designed for simultaneous algorithms, is used in the algorithm to speed up the fully sequential cyclic subgradient projections method. A convergence proof is presented. The advantage of using this strategy is demonstrated with some examples.

We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

We describe a class of stopping rules for Landweber-type iterations for solving linear inverse problems. The class includes both the discrepancy principle (DP rule) and the monotone error rule (ME rule). We also unify the error analysis of the two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using a trained rule are demonstrated on examples taken from image reconstruction from projections. After training the stopping rules became quite robust and only small differences were observed between, e.g. the DP rule and ME rule.

Kaczmarz's method—sometimes referred to as the algebraic reconstruction technique—is an iterative method that is widely used in tomographic imaging due to its favorable semi-convergence properties. Specifically, when applied to a problem with noisy data, during the early iterations it converges very quickly toward a good approximation of the exact solution, and thus produces a regularized solution. While this property is generally accepted and utilized, there is surprisingly little theoretical justification for it. The purpose of this paper is to present insight into the semi-convergence of Kaczmarz's method as well as its projected counterpart (and their block versions). To do this we study how the data errors propagate into the iteration vectors and we derive upper bounds for this noise propagation. Our bounds are compared with numerical results obtained from tomographic imaging.

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from X-ray CT projection data.

We study a class of block-iterative (BI) methods proposed in image reconstruction for solving linear systems. A subclass, symmetric block-iteration (SBI), is derived such that for this subclass both semi-convergence analysis and stopping-rules developed for fully simultaneous iteration apply. Also results on asymptotic convergence are given, e.g., BI exhibit cyclic convergence irrespective of the consistency of the linear system. Further it is shown that the limit points of SBI satisfy a weighted least-squares problem. We also present numerical results obtained using a trained stopping rule on SBI.

This paper is concerned with an inverse problem involving a two-phase moving boundary in two dimensional solidification of pure substance. Using a unique continuation result due to Saut and Schcurer we prove a uniqueness result.

Analysis of a generalized relaxation of string averaging operators, Comment: 20 page