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This paper presents three tensor randomized extended Kaczmarz methods for solving a tensor inconsistent linear system of equations under the t-product between tensors. The randomized extended Kaczmarz, the randomized extended block Kaczmarz, and the randomized extended greedy block Kaczmarz methods in tensor form are proposed to solve inconsistent tensor linear system of equations, respectively. The convergence of each method is proved. Several numerical examples are given to show the efficiency and effectiveness of our methods. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, efficient algorithms will be studied for solving nonlinear saddle point problems from the elliptic optimal control problems. Based on the preconditioned inexact Uzawa (PIU) algorithmic framework, an improved preconditioned inexact Uzawa (I-PIU) algorithmic framework will first be constructed for solving a class of block-structured nonlinear systems, which possess a more general form of the nonlinear saddle point systems. When the I-PIU algorithmic framework is used for solving the block-structured nonlinear systems, a detailed theoretical analysis of the global convergence properties is proposed under the case that the nonlinear operator is single-valued. Besides, a special I-PIU method is established based on a new matrix splitting technique at the first iterative step of the I-PIU algorithmic framework, and then we establish the detailed theoretical results of the convergence. Finally, some numerical experiments are given to demonstrate the efficiency of the I-PIU method. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive framework for studying Krylov subspace methods used to solve the linear system A x = f . These methods aim to achieve convergence within a specified number of iterations, denoted by m, given a particular initial estimate vector x 0 and its corresponding residual r 0 = f - A x 0 . Our analysis focuses on the minimal polynomial Φ m of degree m of A for the vector r 0 . We establish that these methods encompass Petrov-Galerkin methods and minimal seminorm methods as special cases. Additionally, we demonstrate that minimal seminorm methods satisfy implicit Petrov-Galerkin conditions. We provide a general formulation for the iterates based on generalized inverses. The choice of a specific left inverse and the method of constructing the Krylov basis are crucial distinguishing factors among different Krylov subspace methods. We describe and analyze the mathematical properties of these methods, emphasizing their dependency on two matrices. Notably, we prove that CMRH and QMR, as specific instances, also satisfy implicit Petrov-Galerkin orthogonality conditions. Furthermore, we explore techniques to improve the convergence behavior of these methods by carefully selecting vectors in their implementations. Through our investigation, we aim to deepen the understanding of Krylov subspace methods, provide insights into their convergence properties, and identify potential enhancements. We also consider some Krylov subspace methods, which are of product-type methods. In this case, the kth residual r k associated with the approximation x k of the exact solution is given by r k = Ψ k (A) Φ k (A) r 0 , and Ψ k is a polynomial of fixed or variable degree. We will examine particular choices of Ψ k involving local convergence, smoothing, fixed memory, and cost for each iteration. We will also give an enhancement of some product-type methods such as CGS. To illustrate the performance of the derived algorithms, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we intend to utilize the generalized scalar auxiliary variable (GSAV) approach proposed in recent paper (Ju et al., SIAM J. Numer. Anal., 60 (2022), 1905–1931) for the nonlocal coupled sine-Gordon equation to construct a class of fully decoupled, linear, and second-order accurate energy-preserving scheme. The unconditional unique solvability and discrete energy conservation law of the proposed scheme are rigorously discussed, and the unconditional convergence is then proved by the mathematical induction. Particularly, the convergence analysis covers the proposed scheme in multiple dimensions due to the corresponding nonlinear terms satisfy the global Lipschitz continuity straightforwardly. Finally, time evolution of dynamical behavior of the governing equation with different nonlocal parameters are observed, and ample numerical comparisons demonstrate that the proposed scheme manifests high efficiency in long-time computations. [ABSTRACT FROM AUTHOR]

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for one- and two-dimensional Riesz space variable-order fractional diffusion equations. We first introduce fractional centered difference (FCD) and weighted and shifted fractional centered difference (WSFCD) schemes for Riesz space variable-order fractional derivatives. Then the Crank-Nicolson (CN) scheme and the linearly implicit conservative (LIC) difference scheme are applied to discretize the time derivative in linear and nonlinear problems, respectively. Thus, we get CN-FCD and CN-WSFCD schemes, and LIC-FCD and LIC-WSFCD schemes, respectively. Theoretical results about the stability and convergence for the above-mentioned schemes are presented and proved. Banded preconditioners are introduced to speed up GMRES methods for solving the discretization linear systems. The spectral property of the preconditioned matrix is analyzed. Numerical results show that the proposed schemes and preconditioners are very efficient. [ABSTRACT FROM AUTHOR]

Extrapolation, restart and stepsize are very powerful strategies for accelerating the convergence rates of first-order algorithms. In this paper, we propose a modified accelerated proximal gradient algorithm (modAPG), which incorporates the adaptive nonmonotone stepsize strategy, extrapolation and a modified rule of setting the extrapolation parameter to zero at some iterations, for solving a minimization problem composed of a smooth (possibly nonconvex) with Lipschitz continuous gradient and nonsmooth (possibly nonconvex) function. Moreover, the proposed algorithm has a simpler form for this problem with convex setting. We show that any cluster point of the sequence generated by modAPG is a critical point of the problem, and analyze the convergence rates of function values and iterates under the assumption that objective function has the Kurdyka-Łojasiewicz property. Finally, we conduct some preliminary numerical experiments for solving the convex problems, such as image deblurring problem and sparse logistic regression problem; and the nonconvex regularization problem, such as the L 1 2 penalty and the smoothly clipped absolute deviation (SCAD) penalty problem. Numerical experiments demonstrate the promising performance of the proposed method. [ABSTRACT FROM AUTHOR]

In the financial market, the change in price of underlying stock following fractal transmission system is modeled by time-fractional Black-Scholes partial differential equations (PDEs). In this paper, we propose a numerical scheme on uniform mesh for solving time-fractional Black-Scholes PDEs governing European options. The fractional time derivative defined in the Caputo sense is discretized by using the classical L1-scheme and the spatial derivatives are discretized by the cubic spline method. The stability and convergence of the proposed method are analyzed and shown to be second order accurate in space with (2 - α) order accuracy in time. Two numerical examples with exact solutions are presented to verify the efficiency and accuracy of the method validating the theoretical results. Finally, three different types of European options governed by time-fractional Black-Scholes PDE are priced using our proposed method as an application. Further, the impact of the order of time-fractional derivative on the option price is shown. [ABSTRACT FROM AUTHOR]

This paper studies numerical methods for the two-dimensional fractional Gray-Scott (GS) model with fractional Laplacian. A three-level linearized difference scheme for solving the fractional GS model is established. The boundedness, existence, uniqueness and convergence of the finite difference scheme are strictly proved by discrete energy analysis method. Furthermore, the convergence of the difference scheme in l ∞ -norm is of second order in space and time. Numerical experiments are implemented to verify the above theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we construct a stable finite difference scheme for the generalized non-Fickian time-fractional reaction-diffusion equations with time delay. The proposed difference scheme has second-order accuracy in both space and time directions. The stability and convergence of the difference solutions are proved rigorously in the maximum norm. Three representative models with delay are carried out to verify the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, first we propose a general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems. In order to improve the computing efficiency, we further propose a preconditioned general modulus-based matrix splitting iteration method. We establish the convergence theorems when the coefficient matrices are (symmetric) positive definite matrices and H+-matrices, respectively. Numerical results show that the proposed preconditioned general modulus-based matrix splitting iteration method is superior than some existing methods and the general modulus-based matrix splitting iteration method. [ABSTRACT FROM AUTHOR]

An iterative method with fifth order of convergence for solving nonlinear systems is formulated and analyzed. Primary goal of the development of method is to keep the both, convergence order and computational efficiency, as high as possible. Both of these factors are investigated thoroughly in theoretical as well as numerical manner. Comparison with the existing iterative methods of similar order is carried out to examine the performance of method. Numerical testing clearly indicates the high degree of precision and efficiency of the new method. [ABSTRACT FROM AUTHOR]

A generalized quadrature method is studied for Volterra integral equations with highly oscillatory kernels. According to the kernel, a two-point quadrature rule is constructed by Lagrange's identity at first. The error of the quadrature formula is presented as well. Then, it is employed to discretize the highly oscillatory equation without the need to compute moment. For the convergence, the asymptotic order as well as the classical order of the quadrature method for equation is analyzed. It is shown that the method has asymptotic order two and converges with order two as step length decreases. Some numerical examples are conducted to test its efficiency. [ABSTRACT FROM AUTHOR]

In this paper, four accelerated subspace minimization conjugate gradient methods based on 2-regularization or 3-regularization models with different matrices are proposed. The search directions are generated by minimizing a quadratic approximation and different regularization models of the objective function, and the new directions satisfy the sufficient descent condition. In addition, an acceleration technique is used to improve the stepsize. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. By using the Kurdyka-Łojasiewicz property, we analyze the convergence rates of function sequence and iterative sequence without the convexity of objective function. Numerical experiments show that the proposed methods are very effective, especially for CUTEr test problems with no less than 50 dimensions, the best of the new methods is competitive to the latest limited memory conjugate gradient software package CG_DESCENT(6.8) developed by Hager and Zhang (SIAM J. Optim. 23:2150–2168, 24). [ABSTRACT FROM AUTHOR]

In this paper, a bilinear three-point fourth-order compact operator is applied to solve the two-dimensional (2D) Sobolev equation with a Burgers' type nonlinearity. In order to derive a high-order compact difference scheme, an effective reduction technique for the diffusion term is utilized to convert the original high-order evolutionary equation into a low-order system of equations equivalently, which could be discretized combining a classical linear compact operator and the bilinear compact operator. A three-level averaging technique in temporal dimension is embedded, which exports to a linearized compact difference scheme. The proposed compact difference scheme is proved to be uniquely solvable and convergent based on the energy analysis with the error estimate O (τ 2 + h 1 4 + h 2 4) in L2-norm and H1-norm, where h1, h2, and τ denote the spatial and temporal mesh step sizes, respectively. Extensive numerical experiments are carried out to validate the theoretical results on the convergence rates even though the dynamic capillary coefficient tends to zero. Further extending the idea to the 2D Benjamin-Bona-Mahony-Burgers (BBMB) equation is available. The accuracy of the proposed scheme is also demonstrated compared with that of a second-order numerical scheme without the compact technique. [ABSTRACT FROM AUTHOR]

In this paper, by using a modulus-based matrix splitting method as a smoother, a new modulus-based cascadic multigrid method is presented for solving elliptic variational inequality problems. Convergence of the new method is analyzed. Numerical experiments confirm the theoretical analysis and show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, based on the published works by Bai and Tong (J. Univ. Electr. Sci. Tech. China 22:420–424, 1993) and Bai and Huang (J. Univ. Electr. Sci. Tech. China 23:428–432, 1994), we design a class of new modulus-based matrix splitting methods for solving the implicit complementarity problem (ICP) by a new equivalent fixed-point form of the ICP. Some sufficient conditions to guarantee the convergence of the proposed method are presented. Numerical examples are offered to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a one-dimensional two-phase inverse Stefan problem is studied. The free surface is considered unknown here, which is more realistic from the practical point of view. The problem is ill-posed since small errors in the input data can lead to large deviations from the desired solution. To obtain a stable numerical solution, we propose a method based on discrete mollification combined with space marching. We use the integration matrix in Bernstein polynomial basis for the discrete mollification method. Through this method, our numerical integration does not involve any direct function integration and only contains algebraic calculations. Furthermore, the stability and convergence of the process are proved. Finally, the results of this paper are illustrated and examined through some numerical examples. The numerical examples supporting our theoretical analysis are provided and discussed. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a high-order numerical scheme for the two-dimensional time-fractional reaction-subdiffusion equation. The time-fractional derivative of order α, (α ∈ (0,1)) in the model problem is approximated by means of the L2 − 1σ scheme and the space derivatives are discretized by means of a compact alternating direction implicit (ADI) finite difference scheme. Convergence and stability of the method are analyzed. Five numerical examples are provided to demonstrate the applicability and accuracy of the method. It is shown that the method is unconditionally stable and is of O (Δ t 1 + α + h x 4 + h y 4) accuracy, where Δt is the temporal step size and hx and hy are the spatial step sizes. The computed results are in good agreement with the theoretical analysis. Moreover, the comparison with the corresponding results of existing methods demonstrates that our method has advantages in convergence order and error improvement when solving the time-fractional reaction-subdiffusion equation. The CPU time consumed by the proposed method is provided. [ABSTRACT FROM AUTHOR]

Based on the two-sweep modulus-based matrix splitting iteration (TMMS) method for linear complementarity problems developed by Wu and Li (Comput. Appl. Math. 302: 327–339, 2016), a new generalized variant of the TMMS (GTMMS) method is put forth for solving horizontal linear complementarity problems. By analyzing the convergence of the proposed method, we attain its convergence conditions. The results of numerical experiments not only indicate that the convergence of the proposed method is better, but also analyze and provide the relevant factors affecting the convergence. [ABSTRACT FROM AUTHOR]

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions. [ABSTRACT FROM AUTHOR]

An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as β0 < β1 < ⋯ < βM < 1 is the case of study in a bounded Lipschitz domain in ℝ d . The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe's method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Grönwall's inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive modified weak Galerkin (AMWG) method is considered to solve second-order elliptic problem. Under the assumption of a penalty parameter, by showing reliability of error estimator, comparison of solutions and reduction of error estimator, the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops, is proved to be a contraction, namely, the adaptive algorithm is convergent. Numerical experiments are implemented to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, based on the splittings of the coefficient matrix in the PageRank problem, the coupled iteration algorithms are presented for computing PageRank vector. Convergence conditions of the proposed algorithms are analyzed in detail. Furthermore, the choices of the optimal parameters are discussed for some special cases. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

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 FROM AUTHOR]

For the discrete linear system resulted from the considered steady-state space-fractional diffusion equations, we propose an improved scaled HSS (ISHSS) iteration method and discuss its convergence theory. Then, we construct a fast ISHSS (FISHSS) preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods. We discuss the spectral properties of the FISHSS preconditioning matrix. Numerical experiments show the good performance of the FISHSS preconditioner. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ < 2). We first establish the stability of the proposed method and then derive the optimal convergence rate in H1(0,T;L2(Ω))-norm and suboptimal rate in discrete L ∞ (0 , T ; H 0 1 (Ω)) -norm. Furthermore, we discuss the performance of this method when the true solution has singularity at t = 0 and show that optimal convergence rate with respect to H1(0,T;L2(Ω))-norm can still be achieved by using graded temporal grids. Finally, numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Recently, numerous numerical schemes for solving linear time fractional diffusion-wave equations have been developed. However, most of these methods require relatively high smoothness in time and need extensive computational work and large storage due to the nonlocal property of fractional derivatives. In this paper, an efficient scheme and an alternating direction implicit (ADI) scheme are constructed for one-dimensional and two-dimensional nonlinear time fractional diffusion-wave equations based on their equivalent partial integro-differential equations. The proposed methods require weaker smoothness in time compared to the methods based on discretizing fractional derivative directly. They are proved to be unconditionally stable and convergent with first-order of accuracy in time and second order of accuracy in space. Fast implementations of the proposed methods are presented by the sum-of-exponentials (SOE) approximation for the kernel t− 2+α on the interval [τ,T], where 1 < α < 2. Finally, numerical experiments are carried out to illustrate the theoretical results of our direct schemes and demonstrate their powerful computational performances. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate continuous Runge-Kutta methods for solving a class of nonlinear differential-algebraic equations (DAEs) with constant delay. Real-life processes that involve simultaneously time-delay effect and constraints are usually described by delay DAEs. Solving delay DAEs is more complicated than solving non-delay ones since we should focus on both the time-delay and DAE aspects. Recently, we have revisited linear multistep methods and Runge-Kutta methods for a class of nonlinear DAEs (without delay) and shown the advantages of appropriately modified discretizations. In this work, we extend the use of half-explicit Runge-Kutta methods to a similar class of structured strangeness-free DAEs with constant delay. Approximation of solutions at delayed time is obtained by continuous extensions of discrete solution, i.e., continuous output formulas. Convergence analysis for continuous Runge-Kutta methods is presented. It is shown that order reduction that may happen with DAEs is avoided if we discretize an appropriately reformulated delay DAE (DDAE) instead of the original one. Difficulties arising in the implementation are discussed as well. Finally, numerical experiments are given for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we present modified modulus-based synchronous multisplitting iteration methods based on multisplittings of the system matrix for solving the large sparse linear complementarity problems. The proposed methods extend the existing modulus-based synchronous multisplitting iteration methods to a more general case. We establish the convergence theory of these modified modulus-based synchronous multisplitting iteration methods when the system matrix is an H+-matrix. In particular, we investigate the optimal choice of the parameter matrices in theory. Numerical results confirm that the new iteration methods have higher parallel computational efficiency than the existing modulus-based synchronous multisplitting iteration methods. The proposed methods are applied in reconstruction of two-dimensional image data and show the efficiency. [ABSTRACT FROM AUTHOR]

Inthis paper, we study the global uniqueness and solvability of tensor complementarity problems for ℋ + -tensors. We obtain a sufficient condition of the global uniqueness and solvability of tensor complementarity problems for ℋ + -tensors. We present nonlinear dynamical system models for solving the tensor complementarity problem (TCP). We prove that the presented dynamical system models are stable in the sense of Lyapunov stability theory for considering three classes of structured tensors. The computer simulation results further substantiate that the considered dynamical system can be used to solve the TCP. [ABSTRACT FROM AUTHOR]

A fully derivative-free conjugate residual method, using secant condition, is introduced to solve general large-scale nonlinear equations. Under some conditions, global and linear convergence of the proposed method is established by adopting some backtracking type line search. Some numerical results compared with two existing derivative-free methods are reported to show its efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an accelerated double-step scale splitting (ADSS) iteration method for solving complex linear systems. The convergence of the ADSS iteration method is determined under suitable conditions. Also each iteration of ADSS method requires the solution of two linear systems that their coefficient matrices are real symmetric positive definite. We analytically prove that the ADSS iteration method is faster than the DSS iteration method. Moreover, to increase the convergence rate of this method, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and simulation results will be reported to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the implicit complementarity problems. It is proved that the TMMS iteration methods are convergent under certain conditions when the system matrix is either a positive definite matrix or an H+-matrix. Two numerical examples are given to illustrate the effectiveness of the new proposed iteration methods. Numerical results show that the new proposed TMMS iteration methods have better performance than the existing modulus-based relaxation iteration methods for solving the implicit complementarity problems. [ABSTRACT FROM AUTHOR]

The importance of an adequate inner loop starting point (as opposed to a sufficient inner loop stopping rule) is discussed in the context of a numerical optimization algorithm consisting of nested primal-dual proximal-gradient iterations. While the number of inner iterations is fixed in advance, convergence of the whole algorithm is still guaranteed by virtue of a warm-start strategy for the inner loop, showing that inner loop "starting rules" can be just as effective as "stopping rules" for guaranteeing convergence. The algorithm itself is applicable to the numerical solution of convex optimization problems defined by the sum of a differentiable term and two possibly non-differentiable terms. One of the latter terms should take the form of the composition of a linear map and a proximable function, while the differentiable term needs an accessible gradient. The algorithm reduces to the classical proximal gradient algorithm in certain special cases and it also generalizes other existing algorithms. In addition, under some conditions of strong convexity, we show a linear rate of convergence. [ABSTRACT FROM AUTHOR]