Abstract: In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.

Abstract: In this paper, based on the three-term conjugate gradient method and the hybrid technique, we propose a hybrid three-term conjugate gradient projection method by incorporating the adaptive line search for solving large-scale nonlinear monotone equations with convex constraints. The search direction generated by the proposed method is close to the one yielded by the memoryless BFGS method, and has the sufficient descent property and the trust region property independent of line search technique. Under some mild conditions, we establish the global convergence of the proposed method. Our numerical experiments show the effectiveness and robustness of the proposed method in comparison with two existing algorithms in the literature. Moreover, we show applicability and encouraging efficiency of the proposed method by extending it to solve sparse signal restoration and image de-blurring problems.

Abstract: We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (xj). Furthermore, the sequence of local maximum errors per interval (xj− 1,xj) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails.

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

Abstract: 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 polynomial z3 − 1 on a complex plane.

Abstract: We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the cost operators. We prove weak and strong convergence theorems for the sequences generated by these new methods. The numerical behavior of the proposed algorithms when applied to several test problems is compared with that of several previously known algorithms.

Abstract: For solving large-scale consistent systems of linear equations by iterative methods, a fast block Kaczmarz method based on a greedy criterion of the row selections is proposed. The method is deterministic and needs not compute the pseudoinverses of submatrices or solve subsystems. It is proved that the method will converge linearly to the unique least-norm solutions of the linear systems. Numerical experiments are given to illustrate that the method is more efficient and yields a significant acceleration in convergence for the tested data.

Abstract: An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$ ) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → 1−.

Abstract: In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided.

Abstract: In this paper, for solving horizontal linear complementarity problems, a two-step modulus-based matrix splitting iteration method is established. The convergence analysis of the proposed method is presented, including the case of accelerated overrelaxation splitting. Numerical examples are reported to show the efficiency of the proposed method.

Abstract: The paper is focused on analyzing the conservation issues of stochastic 𝜃-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic 𝜃-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.

Abstract: In this paper, we propose several novel numerical techniques to deal with nonlinear terms in gradient flows. These step-by-step solving schemes, termed 3S-SAV and 3S-IEQ schemes, are based on recently popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) approaches. By introducing a novel auxiliary variable η to replace the original one in the traditional SAV approach, we rewrite the equivalent gradient flow systems. Then, linear, decoupled, and unconditionally energy stable numerical schemes are constructed. More importantly, the phase function ϕ and auxiliary variable η can be calculated step-by-step which can save more CPU time in calculation. Similar procedure can also be used to modify the IEQ approach. Specially, the proposed 3S-IEQ approach only needs to solve linear equation with constant coefficients while the system with variable coefficients must be calculated for the traditional IEQ approach. Two comparative studies of traditional SAV/IEQ and 3S-SAV/3S-IEQ approaches are considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

Abstract: We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration of the space-fractional Allen-Cahn equation, which inherits the maximum principle preserving and energy stability. The space-fractional Allen-Cahn equation with homogeneous Dirichlet boundary condition is first discretized in the spatial direction by using a second-order fractional centered difference scheme that preserves the semi-discrete maximum principle. It is subsequently integrated in the temporal direction by a class of strong stability-preserving implicit-explicit Runge-Kutta schemes that are specifically designed to preserve the maximum principle to the optimal time step size. The convergence order in the discrete $L^{\infty }$ norm and energy boundedness are provided by using the established maximum principle. Finally, a series of numerical experiments are carried out to demonstrate the high-order convergence, maximum principle preserving, and energy stability of the proposed schemes.

Abstract: We investigate the techniques and ideas used in Shefi and Teboulle (SIAM J Optim 24(1), 269–297, 2014) in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration, we allow the use of variable metrics, while the investigations are carried out in the setting of infinite-dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties.

Abstract: The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of sketch and project methods. Recently, a block Gaussian version was proposed that uses a block Gaussian sketch, enjoying the regularization properties of Gaussian sketching, combined with the acceleration of the block variants. Theoretical analysis was only provided for the non-block version of the Gaussian sketch method. Here, we provide theoretical guarantees for the block Gaussian Kaczmarz method, proving a number of convergence results showing convergence to the solution exponentially fast in expectation. On the flip side, with this theory and extensive experimental support, we observe that the numerical complexity of each iteration typically makes this method inferior to other iterative projection methods. We highlight only one setting in which it may be advantageous, namely when the regularizing effect is used to reduce variance in the iterates under certain noise models and convergence for some particular matrix constructions.

Abstract: In this paper, based on two-grid discretizations, two kinds of local and parallel finite element methods are proposed and investigated for the coupled Stokes/Darcy model. Following the idea presented in Xu and Zhou (Math. Comput. 69, 881–909 1999). a classical local and parallel finite element method is proposed and investigated. To derive global continuous approximations, a new local and parallel finite element method is devised by combining the partition of unity. We theoretically analyze the resulting formulations and derive optimal error estimates. Numerical experiments are reported to assess the theoretical results.

Abstract: This article deals with an establishment and sharp theoretical analysis of a numerical scheme devised for solving the multi-dimensional multi-term time fractional mixed diffusion and wave equations. The governing equation contains both fractional diffusion term and fractional wave term which make the numerical analysis challenging. With the help of the method of order reduction, we convert the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively, and then develop L2-1σ formula for solving the latter problem. In addition, the formula is used to numerically solve the time distributed-order diffusion and wave equations. The stability and convergence of these numerical schemes are rigorously analyzed by the energy method. The convergence rates are of order two in both time and space. A difference scheme on nonuniform time grids is also constructed for solving the problem with weak regularity at the initial time. Finally, we illustrate our results with some examples.

Abstract: A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered The problem is discretized by the upwind finite-difference method The analysis of this method on Shishkin-type meshes has been well-established, but the discretization mesh in this paper is the original Bakhvalov mesh, introduced in 1969 as the first layer-adapted mesh We analyze the error of the numerical solution in the maximum norm and prove first-order pointwise accuracy, uniform in the perturbation parameter This is the first complete analysis of this kind for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh Our numerical experiments validate the theoretical analysis

Abstract: Two new Runge–Kutta (RK) pairs of orders 6(4) and 7(5) are presented for solving numerically the inhomogeneous linear initial value problems with constant coefficients. These new pairs use only six and eight stages per step respectively. Six stages are needed for conventional Runge–Kutta pairs of orders 5(4) while for such a pair of orders 6(5) we use eight stages. Thus, our proposal is an improvement and it is achieved since the set of order conditions is smaller in the case of interest here. Since traditional simplifications for derivation of Runge–Kutta methods do not apply for this reduced set, we proceed using the differential evolution technique for solving it. We finalize by performing tests over some relevant problems with very pleasant results.

Abstract: In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth- and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application-oriented problems. We provide an implementation of the proposed scheme on nonlinear optimization problem and zero-residual nonlinear least-squares problems taken from the constrained and unconstrained testing environment test set. Finally, based on numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order, efficiency, and computational time and also that the stability results provide the most efficient member of each class of iterative schemes.

Abstract: In this paper, we develop an efficient spectral Galerkin method for the three-dimensional (3D) multi-term time-space fractional diffusion equation. Based on the L2-1σ formula for time stepping and the Legendre-Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed and the stability and convergence analyses are rigorously established. The results show that the fully discrete scheme is unconditionally stable and has second-order accuracy in time and optimal error estimation in space. In addition, we give the detailed implementation and apply the alternating-direction implicit (ADI) method to reduce the computational complexity. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the fractional Bloch-Torrey model is also solved.

Abstract: The ancient concept of circumcenter has recently given birth to the circumcentered-reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection-based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.

Abstract: This paper is devoted to the derivation and analysis of accurate and efficient perfectly matched layers (PMLs) or efficient absorbing layers for solving fractional Laplacian equations within initial boundary value problems (IBVP). Two main approaches are derived: we first propose a Fourier-based pseudospectral method, and then present a real space method based on an efficient computation of the fractional Laplacian with PML. Some numerical experiments and analytical results are proposed along the paper to illustrate the presented methods.

Abstract: This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

Abstract: In this paper, two-grid methods (TGMs) are developed for a system of reaction-diffusion equations of bacterial infection with initial and boundary conditions. The backward Euler (B-E) and Crank–Nicolson (C-N) fully discrete schemes are established, and the existence and uniqueness of the solutions of these schemes are proved. Moreover, based on the combination technique of the interpolation and Ritz projection and derivative transfer trick which are important ingredients in the TGMs, the superclose estimates of order O(h2 + H4 + τ) and O(h2 + H4 + τ2) in H1-norm are deduced for the above schemes, respectively, where h is fine mesh size, H is coarse mesh size, and τ is time step size. Then, by the interpolated postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some other popular finite elements are discussed and numerical results are provided to verify the theoretical analysis, which show that the computing cost of TGMs are only half of Galerkin finite element methods (FEMs) for the test problem.

Abstract: Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their usefulness. This paper develops a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Extensive numerical examples illustrate the performance of the proposed methods.

Abstract: As a tractable approach, regularization is frequently adopted in sparse optimization. This gives rise to regularized optimization, which aims to minimize the l0 norm or its continuous surrogates that characterize the sparsity. From the continuity of surrogates to the discreteness of the l0 norm, the most challenging model is the l0-regularized optimization. There is an impressive body of work on the development of numerical algorithms to overcome this challenge. However, most of the developed methods only ensure that either the (sub)sequence converges to a stationary point from the deterministic optimization perspective or that the distance between each iteration and any given sparse reference point is bounded by an error bound in the sense of probability. In this paper, we develop a Newton-type method for the l0-regularized optimization and prove that the generated sequence converges to a stationary point globally and quadratically under the standard assumptions, theoretically explaining that our method can perform surprisingly well.

Abstract: In this paper, a fast temporal second-order compact ADI scheme is proposed for the 2D time multi-term fractional wave equation. At the super-convergence point, the multi-term Caputo derivative is approximated by combining the order reduction technique with the sum-of-exponential approximation to the kernel function appeared in Caputo derivative. The difference scheme can be solved by the recursion, which reduces the storage and computational cost significantly. The obtained scheme is uniquely solvable. The unconditional convergence and stability of the scheme in the discrete H1-norm are proved by the discrete energy method and the convergence accuracy is second-order in time and fourth-order in space. Numerical example illustrates the efficiency of the scheme.

Abstract: In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and ν-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.