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Abstract: In this paper, we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix. The solver we will derive is based on the Levinson algorithm, which is used for solving strongly nonsingular Toeplitz systems. In the first part an solver for a semiseparable matrix of semiseparability rank p is derived, and in a second part we derive an solver for a band matrix with bandwidth . Both solvers are constructed in a similar way: firstly a Yule–Walker-like equation needs to be solved, and secondly this solution is used for solving a linear equation with an arbitrary right-hand side. Finally, a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices. The overall complexity of this solver is plus lower order terms. In the final section numerical experiments are performed. Attention is paid to the timing and the accuracy of the described methods. [Copyright &y& Elsevier]

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n -dimensional space. [ABSTRACT FROM AUTHOR]

The Peaceman–Rachford splitting method (PRSM) has been widely studied in recent years because of its excellent numerical performance, especially with inertial acceleration. In this paper, we propose a multi-step inertial strictly contractive PRSM (shortly, MISCPRSM) for solving convex optimization problems, which has not been studied before. This paper aims to give an exhaustive analysis of the value relationship between multiple inertial parameters. The second subproblem utilizes an additional customized indefinite proximal term in order to obtain a closed-form solution. The global convergence and complexity result of the introduced method are analyzed by using variational inequality and basic inequalities. Finally, through simulation and computed tomography experiments, the effectiveness and robustness of the MISCPRSM are proved, and the detailed values of inertial parameters are given. [ABSTRACT FROM AUTHOR]

In this paper, an accelerated spectral conjugate gradient projection algorithm is proposed for solving the constrained nonlinear pseudo-monotone equations. We set a restart procedure related to the conjugate parameter in the search direction of the spectral conjugate gradient method. We use an accelerated gradient descent scheme to generate the spectral parameter. Finally, we adopt the relaxed-inertial strategy to accelerated algorithm iteration process. Global convergence is proved under mild conditions without the Lipschitz continuity assumption. Numerical comparisons with other methods show the performances of our algorithm for solving large-scale equations. Our applications are on regularized decentralized logistic regression and image de-blurring problems. [ABSTRACT FROM AUTHOR]

The outbreak of uncertain events, e.g., financial crisis, regional conflict and abrupt contagion, has a significant impact on residents' income. Hence, the wealth management and portfolio selection become more and more important. In addition, the behavioral finance believes that decision-making process of the investors not only depend on utility maximization, but also on who to compare with. It differs from the traditional finance based on rational cognitive acquiring and decision-making of agents, in this paper, we consider both investment uncertainty and investment utility with reference level for portfolio selection. First, we introduce the uncertainty theory and loss aversion utility with reference level based on behavioral finance. Then we study an uncertain multi-period portfolio selection problem. Moreover, we formulate an uncertain behavioral portfolio selection model with three objective functions for which we propose an equivalent program consisting of a main model and two auxiliary models for tri-objective optimization model. Thereafter, we use a modified evolutionary root system growth algorithm to solve the transformed models. Finally, a numerical simulation allows to show the practicability of proposed models and validity of modified algorithm. [ABSTRACT FROM AUTHOR]

Abstract This paper suggests a novel continuous-time gradient descent algorithm of a constrained convex unknown function with a stochastic noise in the observed data. The penalty function approach is employed to introduce the restrictions of the system and to provide a successful optimization process. The solution is restricted to a static scheme dealing with the class of strongly convex functions subject to a set of constraints. To estimate the stochastic gradient we employ a modified version of the synchronous detection method. All the parameters of the proposed approach are decreasing in time to both compensate noise effect in the observations and to provide the mean-square convergence of the suggested approach. We present two different numerical examples to validate the contributions of the paper. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new viscosity iterative method minimization algorithm for image restoration algorithm is proposed. Based on a new viscosity iteration, the algorithm for finding the common zeros of two accretive operators in the framework of uniformly smooth Banach spaces. Moreover, the strong convergence theorems for the iterative algorithms and an example is proposed which shows the validity of main theorem are proved. The results of this paper are improved and extended of the corresponding ones announced by many others and we also applied our result to solve a convex minimization problem and Gâteaux differentiable E P s and Variational Inequality. Experiment results show that the proposed algorithms outperform some other methods. Finally, we give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for applying to solve the image deblurring and image recovery problems. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a simplified reproducing kernel method to solve the linear Volterra integral equations with variable coefficients. The main idea of the method is to establish a reproducing kernel direct space that can be used in Volterra integral equations. And in the first time, this paper analyzes the convergence order and stability of the approximate solution. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than the second order convergence. The algorithm is proved to be feasible and stable through some numerical examples. [ABSTRACT FROM AUTHOR]

In the paper the author presents a novel kind of the WZ factorization algorithm, namely a block WZ factorization algorithm. The aim of this new algorithm is to utilize the computational power of contemporary computers with hierarchical memory. In the paper, some properties of the matrix Z are given and analyzed. Next, a version of the block WZ factorization is presented. The author shows that such a block WZ factorization exists for strictly diagonally dominant matrices. The computational cost of this block algorithm is presented. The time and the accuracy of proposed block WZ factorization algorithm for random dense square diagonally dominant matrices are reported. The block algorithm turned out to be faster even up to 300 times than the original WZ factorization. [ABSTRACT FROM AUTHOR]

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them. [ABSTRACT FROM AUTHOR]

We study a multi-block separable convex optimization problem with the linear constraints, where the objective function is the sum of m individual convex functions without overlapping variables. The linearized version of the generalized alternating direction method of multipliers (L-GADMM) is particularly efficient for the two-block separable convex programming problem and its convergence was proved when two blocks of variables are alternatively updated. However, the convergence analysis and practical applications of the extended L-GADMM (m ≥ 3) are in their early stages. In this paper, we extend this algorithm to the general case where the objective function consists of the sum of m -block convex functions. Theoretically, we prove global convergence of the new method and establish the worst-case convergence rate in the ergodic and nonergodic senses for the proposed algorithm. The efficiency of the new method is further demonstrated through numerical results on the calibration of the correlation matrices. [ABSTRACT FROM AUTHOR]

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

Box-constrained optimization problems in the real m × n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for block sparse signal recovery that minimizes the block q -ratio sparsity ‖ z ‖ 2 , 1 / ‖ z ‖ 2 , q q q − 1 with q ∈ [ 0 , ∞ ]. For the case of 1 < q ≤ ∞ , we present the theoretical analyses and the computing algorithms for both cases of the ℓ 2 -bounded and ℓ 2 , ∞ -bounded noises. The corresponding unconstrained model is also investigated. Its superior performance in block sparse signal reconstruction is demonstrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a new descent approximate modified residual algorithm is developed to solve a large scale system of nonlinear symmetric equations, where the basic strategy to improve its numerical performance is to approximately compute the gradients and the difference of gradients. The error bounds of this approximation are presented, and in virtue of this approximation, a conjugate gradient algorithm for solving large-scale optimization problems in the literature is extended to solve the system of nonlinear symmetric equations without needs of computing and storing the Jacobian matrices or their approximate matrices. It is proved that the obtained search directions in our developed algorithm are sufficiently decent with respect to the so-called approximate modified residues. Under mild assumptions, global and local convergence results of the developed algorithm are proved. Numerical tests indicate that the developed algorithm outperforms the other similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a generalized version of proximal method with a new step size update for solving variational inequality problem in Hilbert spaces. Weak convergence of the proposed scheme is attained under some appropriate assumptions imposed on the operators and parameters. At the same time, we set up an R-linear convergence rate of our method when the operator is strong monotone. Numerical experiments are given to demonstrate the efficacy of our proposed iterative algorithm. Several versions of recently proposed methods in the literature are special cases of our method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new self-adaptive iterative algorithms for finding the solution of a split variational inclusion problem in real Hilbert spaces. The strong convergence theorems are proved under suitable conditions. In applications, the main results are applied for solving the split feasibility problem, the split minimization problem, and the split common fixed point problem. In order to test the numerical performances, the proposed algorithms are compared with several existing algorithms in the literature. Finally, the computational experiments show that the proposed algorithms are fast and more efficient than other approaches. [ABSTRACT FROM AUTHOR]

The use of decision rules allows to extract information and to infer conclusions from relational databases in a reliable way, thanks to some indicators like support and certainty. Moreover, decision algorithms collect a group of decision rules that satisfies desirable properties to describe the relational system. However, when a decision table is considered within a fuzzy environment, it is necessary to extend all notions related to decision algorithms to this framework. This paper presents a generalization of these notions, highlighting the new definitions of indicators of relevance to describe decision rules and decision algorithms. [ABSTRACT FROM AUTHOR]

Finite tight frames are used widely for many applications. An important problem is to construct finite tight frames containing specified elements as needed in real world applications. The question was first studied in details by D. Feng, L. Wang, and Y. Wang in 2007. P. G. Casazza, M. Ficus, D. G. Mixon, J. Peterson, and I. Smalyanau gave a method for constructing a tight frame containing specified elements in 2013. A simple and stable method for constructing equi-norm tight frames from the specified elements was derived by J. Ji and L. Wang in 2022. As the main result, an algorithm based on the QR factorization for completing a Mercedes-Benz frame containing specified elements is presented in this paper and two examples of Mercedes-Benz frames in R 4 are constructed to illustrate our approach. [ABSTRACT FROM AUTHOR]

In this paper, we study stochastic composite problems where the objective can be the composition of an outer single-valued function and an inner vector-valued mapping. In this stochastic composite optimization, the inner mapping can be expressed as an expectation over random component mappings. In this study, we propose two algorithms to address the generality and possible singularities of this problem and bound their sample complexities for finding an ϵ -stationary point. The first algorithm is the prox-linear hybrid stochastic gradient algorithm, which may achieve sample complexities of O (ϵ 2 τ − 5 / 2) and O (ϵ τ − 3 / 2) for the component mappings and their Jacobians respectively, where τ ∈ [ 0 , 1 ]. The second algorithm is the normalized proximal hybrid stochastic gradient algorithm, which takes advantage of the special structure of the regularizer. This algorithm may achieve sample complexities of O (ϵ 2 τ − 4) for both the component mappings and the Jacobians, where τ ∈ [ 5 / 4 , 7 / 4 ]. Numerical experiments prove that the two proposed algorithms are quite competitive with other existing algorithms. A real-life application in sparse portfolio selection problems is also promising. • We study a class of stochastic composite optimization problems. • We present two hybrid SGD algorithms for the considered problem. • We investigate the convergence rates and complexity of the algorithms. • We apply the proposed approach to the sparse portfolio selection problem. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam’s method with that of this paper. [ABSTRACT FROM AUTHOR]

This paper introduces a hybrid algorithm of Nelder–Mead and scatter search algorithms called SSNM. Numerical results for some known test problems with varying dimensions from 2 to 100 variables prove its great performance for a bounded or constrained problem. This paper also has a considerable contribution to the joint economic lot sizing (JELS) problem literature. This problem is considered for a two-stage supply chain with price-sensitive demand for geometric, geometric-then-equal size and optimal shipment policies. Solution procedures are also developed which use the SSNM algorithm in their steps. These models and solution procedures are novel in the JELS literature. [ABSTRACT FROM AUTHOR]

To find new solutions through the information provided by previously found solution(s), this paper is first to develop a new singular transform to change the local basin-barrier structure of the original problem in an infinite-dimensional space and then to propose an augmented partial Newton method to solve the transformed problem on the solution set. Mathematical justifications of the new formulation and method are established. Details on algorithm implementation are presented. Examples for two very different variational PDE problems are carried out to illustrate the method. Their numerical results are displayed by profile–contour plots and error/convergence data. The formulation and method developed in this paper can also be applied to solve multiple fixed point problems. [ABSTRACT FROM AUTHOR]

This paper presents an accurate implicit re-initialization approach in the framework of a level set method. The improved method includes two schemes for keeping the zero level set unperturbed. The first scheme is to derive and use a new formula for the smoothing parameter in the conventional re-initialization equation based on the principle of the interface not moving. The second scheme is to reduce the local time step to avoid the interface moving across grid points when the sign of the level set function near the interface is changed. The example presented suggests that the new algorithm has a better approximation to the signed distance function and obtains a more accurate interface than the algorithm presented by Sussman et al. (1994). For shape and topology optimization of fluid, the effectiveness of the developed method in this paper was demonstrated by the internal and external flow examples. [ABSTRACT FROM AUTHOR]

In this paper new algorithms are being proposed for evaluation of highly oscillatory integrals (HOIs) with stationary point(s). The algorithms are based on modified Levin quadrature (MLQ) with multiquadric radial basis functions (RBFs) coupled with quadrature rules based on hybrid functions of order 8 (HFQ8) and Haar wavelets quadrature (HWQ) (Aziz et al. 2011). Part of the new procedure presented in this paper is comprised of transplanting monomials (which are used in the conventional Levin method) by the RBFs. The linear and Hermite polynomials based quadratures (Xiang, 2007) are being replaced by the new methods based on HWQ and HFQ8 respectively. Both the methods are merged with MLQ to obtain the numerical solution of highly oscillatory integrals having stationary points. The accuracy of the new methods is neither dampened by presence of the stationary point(s) nor by the large value of frequency parameter ω . Theoretical facts about the error analysis of the new methods are analyzed and proved. Numerical examples are included to show efficiency and accuracy of the new methods. [ABSTRACT FROM AUTHOR]

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

In this paper, using the properties of graph convergence of H(.,.)-accretive operators, we construct a perturbed algorithm for solving systems of ill-posed variational inclusions involving H(.,.)-accretive operators. Under some suitable conditions, we prove that the sequence generated by the proposed method is strongly convergent to the unique solution of the problem. The main results presented in this paper generalize and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach. [Copyright &y& Elsevier]

Abstract: Theoretical analyses of algorithms are important to understand their search behaviors and develop more efficient algorithms. Compared with the plethora of works concerning the empirical study of the differential evolution (DE), little theoretical research has been done to investigate the convergence properties of DE so far. This paper focuses on theoretical researches on the convergence of DE and presents a convergent DE algorithm. First of all, it is proved that the classical DE cannot converge to the global optimal set with probability 1 by using the property that it cannot escape from a local optimal set. Inspired by the characteristics of the elitist genetic algorithm, this paper proposed a modified DE to overcome the disadvantage. The proposed algorithm employs two operators that assist it in escaping from a local optimal set and enhance the diversity of the population. And it is then verified that the proposed algorithm is capable of converging to global optima with probability 1. The theoretical research of this paper is undertaken in a finite discrete set, and the analysis tool used is the Markov chain. The numerical experiments are conducted on a deceptive function and a set of benchmark functions. The experimental results support the theoretical analyses on the convergence performances of the classical and modified DE algorithm. [Copyright &y& Elsevier]

Powell (1984) and Dai (2003) constructed respectively a counterexample to show that the Polak–Ribière–Polyak (PRP) conjugate gradient algorithm fails to globally converge for nonconvex functions even when the exact line search technique is used, which implies similar failure of the weak Wolfe–Powell (WWP) inexact line search technique. Does another inexact line search technique exist that can ensure global convergence for nonconvex functions? This paper gives a positive answer to this question and proposes a modified WWP (MWWP) inexact line search technique. An algorithm is presented that uses the MWWP inexact line search technique; the next point x k + 1 generated by the PRP formula is accepted if a positive condition holds, and otherwise, x k + 1 is defined by a technique for projection onto a parabolic surface. The proposed PRP conjugate gradient algorithm is shown to possess global convergence under inexact line search for nonconvex functions. Numerical performance of the proposed algorithm is shown to be competitive with those of similar algorithms. • A convex parabolic surface for projection is defined and a projection PRP algorithm is proposed. • Any unsatisfactory iteration points are projected onto this convex parabolic surface to get good points. • The PRP conjugate gradient algorithm for general functions under inexact line search is obtained. • Competitive numerical results are reported to show the good performance of the new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence of several optimize-then-discretize and discretize-then-optimize algorithms, based on either a second-order or a fourth-order finite difference discretization, for solving elliptic PDE-constrained optimal control problems. To ensure the convergence of a discretize-then-optimize algorithm, one well-accepted criterion is to design the discretization scheme such that the resulting discretize-then-optimize algorithm commutes with the corresponding optimize-then-discretize algorithm. In other words, both algorithms should give rise to exactly the same discrete optimality system. However, such a restrictive criterion is not trivial to fulfill. By investigating a distributed control problem governed by an elliptic equation, we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence. We then introduce some suitable H 1 semi-norm penalty/regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity. Numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques. [ABSTRACT FROM AUTHOR]

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

In this paper an improved version of a verification algorithm for solving nonlinear systems of equations based on Krawczyk operator is presented. Compared with the original algorithm, the improved verification algorithm not only saves computing time, but also computes a narrower (or at least the same) inclusion of the solution to nonlinear systems of equations for certain classes of problems. Numerical results demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The reduction of a matrix to an upper J -Hessenberg form is a crucial step in the S R -algorithm (which is a Q R -like algorithm), structure-preserving, for computing eigenvalues and vectors, for a class of structured matrices. This reduction may be handled via the algorithm JHESS or via the recent algorithm JHMSH and its variants. The main drawback of JHESS (or JHMSH) is that it may suffer from a fatal breakdown, causing a brutal stop of the computations and hence, the S R -algorithm does not run. JHESS may also encounter near-breakdowns, source of serious numerical instability. In this paper, we focus on these aspects. We first bring light on the necessary and sufficient condition for the existence of the S R -decomposition, which is intimately linked to J -Hessenberg reduction. Then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns. Hence, the J -Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns. [ABSTRACT FROM AUTHOR]

Image demosaicking and denoising are the first two key steps of the color image production pipeline. The classical processing sequence has for a long time consisted of applying denoising first, and then demosaicking. Applying the operations in this order leads to oversmoothing and checkerboard effects. Yet, it was difficult to change this order, because once the image is demosaicked, the statistical properties of the noise are dramatically changed and hard to handle by traditional denoising models. In this paper, we address this problem by a hybrid machine learning method. We invert the traditional color filter array (CFA) processing pipeline by first demosaicking and then denoising. Our demosaicking algorithm, trained on noiseless images, combines a traditional method and a residual convolutional neural network (CNN). This first stage retains all known information, which is the key point to obtain faithful final results. The noisy demosaicked image is then passed through a second CNN restoring a noiseless full-color image. This pipeline order completely avoids checkerboard effects and restores fine image detail. Although CNNs can be trained to solve jointly demosaicking–denoising end-to-end, we find that this two-stage training performs better and is less prone to failure. It is shown experimentally to improve on the state of the art, both quantitatively and in terms of visual quality. [ABSTRACT FROM AUTHOR]

The rational fraction system is a special nonlinear system, the existence of the denominator polynomial leads to the difficulty of identifying rational fraction models. Inspired by the gradient search and the Newton method, the gradient-based iterative algorithm and Newton iterative algorithm are presented to estimate the parameters of rational fraction system models. Furthermore, in order to avoid a large amount of calculation and complex equations encountered in the process of solving partial derivatives, the model transformation-based gradient iterative algorithm and the model transformation-based Newton iterative algorithm are proposed for parameter identification. Two examples are carried out to show the effectiveness of the proposed algorithms. This paper focuses on solving the identification problem of rational fraction systems. [ABSTRACT FROM AUTHOR]

NURBS method is the standard mathematical method for describing the shapes of curves/surfaces/volumes, and it is extensively used in computer-aided design, computer-aided manufacturing, and computer graphics. The injectivity of NURBS volumes means that they do not have self-intersections. Since the injectivity of parameterizations depend on the signs of their Jacobian functions, and the Jacobian of a NURBS volume is determined by the determinant of its tangent vectors in three directions, we first propose a method to compute the bounding vectors of the tangent cones of NURBS volume in this paper. Then the sufficient condition for the injectivity of NURBS volume is proposed. A checking algorithm is also presented. Some examples are given to verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The paper presents two-scale numerical algorithms for stress–strain analysis of porous media featured by self-contact at pore level. The porosity is constituted as a periodic lattice generated by a representative cell consisting of elastic skeleton and a void pore. Unilateral frictionless contact is considered between opposing surfaces of the pore. For the homogenized model derived in our previous work, we justify incremental formulations and propose several variants of two-scale algorithms which commute iteratively solving of the micro- and the macro-level contact subproblems. A dual formulation which takes advantage of the assumed microstructure periodicity and a small deformation framework, is derived for the contact problems at the micro-level. This enables to apply the semi-smooth Newton method. For the global, macrolevel step two alternatives are tested; one relying on a frozen contact identified at the microlevel, the other based on a reduced contact associated with boundaries of contact sets. Numerical examples of 2D deforming structures are presented as a proof of the concept. • A new two-scale variational method for unilateral self-contact in porous media is derived. • The proposed incremental formulation uses a consistent homogenized elastic tensor. • The new macroscopic contact method couples micro- and macro-scopic contact problems. • A dual formulation is proposed to solve local contact microproblems efficiently. • The presented model and solution method are easily extensible to fluid-filled pores. [ABSTRACT FROM AUTHOR]

The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a k -sparse signal if the restricted isometry constant of the measurement matrix satisfies δ 3 k < 0. 618 and δ 3 k < 0. 577 , respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions δ 2 k < 0. 356 and δ 2 k < 0. 377 , respectively. Moreover, the finite convergence of HBHTP and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a Lorentzian- ℓ p (0 < p < 1) optimization for recovering sparse signals from measurements corrupted by two types of impulsive noise. A smoothing strategy for nonconvex ℓ p norm is employed and an efficient Lorentzian based iterative algorithm is developed for solving the resulting optimization problem. We describe the detailed process of the algorithm and establish the convergence of the proposed algorithm. Numerical results show that our proposed algorithm performs effectively in recovering sparse signals from impulsive measurement noise compared with the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a new parallel accurate algorithm called PAccSumK for computing summation of floating-point numbers. It is based on AccSumK algorithm. In the experiment, for the summation problems with large condition numbers, our algorithm outperforms the PSumK algorithm in terms of accuracy and computing time. The reason is that our algorithm is based on a more accurate algorithm called AccSumK algorithm compared to the SumL algorithm used in PSumK. The proposed parallel algorithm in this paper is designed to compute a result as if computed internally in K -fold the working precision. Numerical results are presented showing the performance and the accuracy of our new parallel algorithm for calculating summation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an effective method based upon the piecewise homotopy perturbation method (PHPM) is proposed for finding the numerical solutions of Fredholm integro-differential equations with weakly singular kernels. For these equations, the traditional homotopy perturbation method is divergent. So we present a modified homotopy perturbation method for solving this kind of equation. The advantages of the approach of this paper lie in the feature that this technique not only achieves an approximate solution with high accuracy, but also improves the calculation speed. Two examples are given to show the effectiveness and convenience of this method. [Copyright &y& Elsevier]

Most algorithms for determining a Jordan basis for an endomorphism of a finite dimensional vector space suffer from the major drawback that they are computationally inefficient. In this paper, a universal and efficient algorithm for an endomorphism of a finite dimensional vector space over an arbitrary field is presented. Three computational examples are considered which show how our new algorithm works. A computational comparison to the Jordan basis algorithm in Kudo et al. (2010) completes the paper (algorithmic aspects). [ABSTRACT FROM AUTHOR]

This paper proposes a class of randomized Kaczmarz algorithms for obtaining isolated solutions of large-scale well-posed or overdetermined nonlinear systems of equations. This type of algorithm improves the classic Newton method. Each iteration only needs to calculate one row of the Jacobian instead of the entire matrix, which greatly reduces the amount of calculation and storage. Therefore, these algorithms are called matrix-free algorithms. According to the different probability selection patterns of choosing a row of the Jacobian matrix, the nonlinear Kaczmarz (NK) algorithm, the nonlinear randomized Kaczmarz (NRK) algorithm and the nonlinear uniformly randomized Kaczmarz (NURK) algorithm are proposed. In addition, the NURK algorithm is similar to the stochastic gradient descent (SGD) algorithm in nonlinear optimization problems. The only difference is the choice of step size. In the case of noise-free data, theoretical analysis and the results of numerical based on the classical tangential cone conditions show that the algorithms proposed in this paper are superior to the SGD algorithm in terms of iterations and calculation time. [ABSTRACT FROM AUTHOR]

Abstract: In the framework of dependent risks it is a crucial task for risk management purposes to quantify the probability that the aggregated risk exceeds some large value . Motivated by Asmussen et al. (2011) [1] in this paper we introduce a modified Asmussen–Kroese estimator for simulation of the rare event that the aggregated risk exceeds . We show that in the framework of log-Gaussian risks our novel estimator has the best possible performance i.e., it has asymptotically vanishing relative error. For the more general class of log-elliptical risks with marginal distributions in the Gumbel max-domain of attraction we propose a modified Rojas-Nandayapa estimator of the rare events of interest, which for specific importance sampling densities has a good logarithmic performance. Our numerical results presented in this paper demonstrate the excellent performance of our novel Asmussen–Kroese algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a new curve subdivision algorithm called interproximate subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By the interproximate subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of continuity. The interproximate subdivision also contains tension parameters assigned to vertices or edges for shape adjustment. Compared to the 4-point interpolatory subdivision scheme, the interproximate subdivision does not force the new inserted vertices to be interpolated and is thus expected to have improved behavior; and compared to the cubic -spline refinement scheme, the interproximate subdivision is able to generate curves interpolating user-specified vertices. In addition, the paper also presents two extensions of the interproximate subdivision: one automatically adapts the tension parameters locally according to the geometry of the control polygon during the refinement to achieve convexity preservation and the other automatically relaxes the interpolating property of some vertices to achieve better shape behavior. [Copyright &y& Elsevier]

Abstract: This paper reports on improvements to recent work on the computation of a structured low rank approximation of the Sylvester resultant matrix of two inexact polynomials and . Specifically, it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of is computed. The existing algorithm may still, however, yield a structured low rank approximation of , but not a structured low rank approximation of , which is unsatisfactory. Moreover, a structured low rank approximation of must be equal to, apart from permutations of its columns, a structured low rank approximation of , but the existing algorithm does not guarantee the satisfaction of this condition. This paper addresses these issues by modifying the existing algorithm, such that these deficiencies are overcome. Examples that illustrate these improvements are shown. [Copyright &y& Elsevier]