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In this paper, we present several remarks on the paper by Yao et al. (citeyearcite.nine). The results presented in the present paper are interesting improvements on the main results of Yao et al.

In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results.

In this paper a finite element analysis to approximate the solution of an obstacle problem for a static shallow shell confined in a half space is presented. First, we rigorously prove some estimates for a suitable enriching operator connecting Morley's triangle to Hsieh-Clough-Tocher triangle. Secondly, we establish an estimate for the approximate bilinear form associated with the problem under consideration. Finally, we conduct an error analysis and we prove the convergence of the proposed numerical scheme.

This short paper concludes a general waveform relaxation (WR) method at the PDE level for semi-linear reaction-diffusion equations. For the case of multiple coupled PDE(s), new Jacobi WR and Gauss-Seidel WR are provided to accelerate the convergence result of classical WR. The convergence conditions are proved based on energy estimate. Numerical experiments are demonstrated with several WR methods in parallel to verify the effectiveness of the general WR method.

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper, we propose a somewhat different approach, in the sense that the algorithm itself is used in order to formulate a new optimality condition satisfied by its feasible limit points. With this tool at hand, we present several new properties and insights on limit points of augmented Lagrangian schemes, in particular, characterizing the strongest possible global convergence result for the safeguarded augmented Lagrangian method.

This paper presents an inertial Tseng extragradient method for approximating a solution of the variational inequality problem. The proposed method uses a single projection onto a half space which can be easily evaluated. The method considered in this paper does not require the knowledge of the Lipschitz constant as it uses variable stepsizes from step to step which are updated over each iteration by a simple calculation. We prove a strong convergence theorem of the sequence generated by this method to a solution of the variational inequality problem in the framework of a 2-uniformly convex Banach space which is also uniformly smooth. Furthermore, we report some numerical experiments to illustrate the performance of this method. Our result extends and unifies corresponding results in this direction in the literature.

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

Consider Laplace’s equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index.

The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.

The purpose of this paper is to investigate the connections between the weak Galerkin (WG) methods with and without stabilizers. The choices of stabilizers directly affect the convergence rates of the corresponding WG methods in general. However, we observed that the convergence rates are independent of the choices of stabilizers for these WG elements with stabilizers being optional. In this paper, we will verify such phenomena theoretically as well as numerically.

In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.

This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented.

In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations.

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular ${\mathscr{M}}$ -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method.

The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces. It is proved that the sequence generated by the proposed iterative algorithm converges strongly to a solution of the split common fixed point problem. Then, the main result is used to study the split common null point problem and the split quasi-inclusion problem. Finally, a numerical example is provided to illustrate our main result. The results presented in this paper extend and improve some recent corresponding results.

Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with ${\mathcal O}(\bar {S}N\log N+\bar {S}^{2} N)$ operation cost on each time level with adaptability analysis, where $\bar {S}$ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method.

This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type $u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })$ , x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.

Zeroing dynamics (ZD) has shown great performance to solve various time-varying problems. In this paper, the problem of time-varying Sylvester-transpose matrix inequality is first investigated. Since it is difficult to solve a matrix inequality with a matrix variable and its transpose by traditional methods, this paper proposes a continuous ZD (CZD) model by employing ZD design formula and JMP function array to solve this challenging problem. Furthermore, for better implementation on digital computers, three discrete ZD (DZD) models are proposed by using three different discretization formulas with different precision, i.e., the Euler-forward formula, the 6-instant Zhang et al discretization (ZeaD) formula and the 7-instant ZeaD formula. What is more, theoretical truncation error analyses and numerical experiments substantiate the convergence, efficacy and superiority of the DZD models for solving time-varying Sylvester-transpose matrix inequality.

The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomialz3− 1 on a complex plane.

There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005).

In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution ${u^{n}_{h}}$ by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of ${u_{h}^{n}}$ and the second-order H1-convergence of the interpolation post-processing function on ${u_{h}^{n}}$ with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.

In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.

It is well-known that conjugate gradient algorithms are widely applied in many practical fields, for instance, engineering problems and finance models, as they are straightforward and characterized by a simple structure and low storage. However, challenging problems remain, such as the convergence of the PRP algorithms for nonconvexity under an inexact line search, obtaining a sufficient descent for all conjugate gradient methods, and other theory properties regarding global convergence and the trust region feature for nonconvex functions. This paper studies family conjugate gradient formulas based on the six classic formulas, PRP, HS, CD, FR, LS, and DY, where the family conjugate gradient algorithms have better theory properties than those of the formulas by themselves. Furthermore, this technique of the presented conjugate gradient formulas can be extended to any two-term conjugate gradient formula. This paper designs family conjugate gradient algorithms for nonconvex functions, which have the following features without other conditions: (i) the sufficient descent property holds, (ii) the trust region feature is true, and (iii) the global convergence holds under normal assumptions. Numerical results show that the given conjugate gradient algorithms are competitive with those of normal methods.

In this paper, we study an inverse source problem for the Helmholtz equation from measurements. The purpose of this paper is to reconstruct the salient features of the hidden sources within a body. We propose three stable reconstruction algorithms to detect the number, the location, the size, and the shape of the hidden sources along with compact support from a single measurement of near-field Cauchy data on the external boundary. This problem is nonlinear and ill-posed; thus, we should consider regularization techniques in reconstruction algorithms. We give several numerical experiments to demonstrate the viability of our proposed reconstruction algorithms.

In a number of our previous papers, we have proposed interval versions of multistep methods (explicit and implicit), including interval predictor-corrector methods, in which the step size was constant. In this paper, we present interval versions of Adams-Bashforth methods with a possibility to change step sizes. This possibility can be used to obtain interval enclosures of the exact solution with a width given beforehand.

In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrodinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.