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Abstract: The Prediction Error Method (PEM) is related to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and/or the search space is constrained to a nonconvex set. The shape of the cost function, and hence the difficulty in solving the optimization problem, depends directly on the experimental conditions, more specifically on the spectrum of the input/output data collected from the system. Therefore, it seems plausible to improve the convergence to the global minimum by properly choosing the spectrum of the input; in this paper, we address this problem. We present a condition for convergence to the global minimum of the cost function and propose its inclusion in the input design. We present the application of the proposed approach to case studies where the algorithms tend to get trapped in nonglobal minima. [Copyright &y& Elsevier]

Abstract: In this paper, the steepest descent algorithm without line search is proposed for p-Laplacian. Its search direction is the weighted preconditioned steepest descent one, and step length is estimated by a formula except the first iteration. Continuation method is applied for solving the p-Laplacian with very large p. Lots of numerical experiments are carried out on these algorithms. All numerical results show the algorithm without line search can cut down some computational time. Fast convergence of these new algorithms is displayed by their step length figures. These figures show that if search direction is the steepest descent one, exact step lengths can be substituted properly with step lengths obtained by the formula. [Copyright &y& Elsevier]

This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem. [ABSTRACT FROM PUBLISHER]

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported. [ABSTRACT FROM PUBLISHER]

In this paper, based on a new class of asymptotic l 1 exact penalty functions, we propose a smooth penalty function for solving nonlinear programming problems. One of the main features of our algorithm is that at each iteration, we do not need to solve the global minimum of penalty functions. Furthermore, global convergence property is established without requiring any constraint qualifications. By addressing perturbation functions, we obtain that the lower semi-continuity of the perturbation function at zero is a necessary and sufficient condition to ensure the convergence of the objective function values generated by the algorithm to the optimal value of the primal problem. Since the perturbation function is only dependent on the data of the primal problem, these results enable us to check the convergence property of the algorithm in advance. Finally, we discuss the finite termination of the algorithm when the solution set is non-degenerate or weakly sharp. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a trust-region sequential quadratic programming algorithm with a modified filter acceptance mechanism is proposed for nonlinear equality constrained optimization. The most important advantage of the proposed algorithm is its avoidance of any feasibility restoration phase, a necessity in traditional filter methods. We solve quadratic programming subproblems based on the well-known Byrd–Omojokun trust-region method. Inexact solutions to these subproblems are allowed. Under some standard assumptions, global convergence of the proposed algorithm is established. Numerical results show our approach is potentially useful. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new partial splitting augmented Lagrangian method for solving the separable constrained convex programming problem where the objective function is the sum of three separable convex functions and the constraint set is also separable into three parts. The proposed algorithm combines the alternating direction method (ADM) and parallel splitting augmented Lagrangian method (PSALM), where two operators are handled by a parallel method, while the third operator and the former two are dealt with by an alternating manner. Under mild conditions, we prove the global convergence of the new method. We also report some preliminary numerical results on constrained matrix optimization problem, illustrating the advantage of the new algorithm over the most recently PADALM of Peng and Wu (2010) [12]. [Copyright &y& Elsevier]

The generalized Nash equilibriumproblem(GNEP) is anextensionof the standard Nash equilibrium problem(NEP), inwhich each player's strategy set may depend on the rival player's strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP. [ABSTRACT FROM AUTHOR]

The job shop scheduling problem (JSSP) is a notoriously difficult problem in combinatorial optimization. Up till now, neighbourhood search has been the most efficient optimization framework for solving the problem. The discovery and innovative application of neighbourhood properties is a key issue in the design of such algorithms. Effective neighbourhood properties help one to exclude unpromising solutions in the neighbourhood, and thus accelerate the convergence of the whole searching process. In this paper, we aim at solving the JSSP with the objective of minimizing total weighted tardiness, which includes most other regular objective functions as special cases. A neighbourhood property of the problem is discovered based on the duality of linear programming models. Then, the neighbourhood property is utilized by a tree search procedure, which is finally embedded into the particle swarm optimization framework. According to extensive computational experiments and the real-world application to a large-scale manufacturing firm in China, the proposed approach is effective for solving the JSSP. [ABSTRACT FROM PUBLISHER]

The main purpose of this paper is to study the split fixed point and equilibrium problems which includes fixed point problems, equilibrium problems, and variational inequality problems as special cases. A damped algorithm is presented for solving this split common problem. Strong convergence analysis is shown. [ABSTRACT FROM AUTHOR]

This paper deals with design algorithms for the split variational inequality and equilibriumproblems. Strong convergence theorems are demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new algorithm is proposed to solve the nonlinear constrained optimization problems. Unlike sequential quadratic programming (SQP) type methods, this algorithm does not involve solutions of quadratic programs. It is merely necessary to solve systems of linear equations with small scale to obtain a direction. The scheme is based on an idea of -effective active set strategies. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions. [Copyright &y& Elsevier]

In this paper, the constitutive relations of the finite element method are constructed and used for solving two- and three-dimensional problems of the asymmetric theory of elasticity. Different variants of finite elements are considered. The numerical experiments are carried out to evaluate the reliability and computational efficiency of the finite element algorithm based on the comparison between the numerical and analytical solutions, numerical estimation of the convergence and checking of the degree of accuracy, to which the natural boundary conditions are satisfied. The obtained solutions to the two- and three-dimensional problems are interpreted from the viewpoint of their applicability to a design of experiments capable of revealing the facts of couple-stress effects in material under elastic deformation and identification of material constants for the asymmetric theory of elasticity. The capabilities of the finite element algorithm to interpret experimental data and estimate the errors occurring in real experiments have been tested by solving several example problems. [ABSTRACT FROM AUTHOR]

The impact of this paper is twofold. First, we study convergence rates of the iteratively regularized Gauss-Newton (IRGN) algorithm with a linear penalty term under a generalized source assumption and show how the regularizing properties of new iterations depend on the solution smoothness. Secondly, we introduce an adaptive IRGN procedure, which is investigated under a relaxed smoothness condition. The introduction and analysis of a more general penalty term are of great importance since, apart from bringing stability to the numerical scheme designed for solving a large class of applied inverse problems, it allows us to incorporate various types of a priori information available on the model. Both a priori and a posteriori stopping rules are investigated. For the a priori stopping rule, optimal convergence rates are derived. A numerical example illustrating convergence rates is considered [ABSTRACT FROM AUTHOR]

In this paper, we will show how to obtain asymptotic solutions for the problem of pricing Asian options. Under the assumption that the underlying follows geometric Brownian motion, we will derive Taylor expansion series for the fixed and floating strike Asian options. While there will be no analytical formulae for calculating expansion coefficients, we will provide relatively simple algorithms for calculating them. The methodology is particularly effective for the case of continuously sampled fixed-strike Asian calls where it takes only seconds to obtain constants for the Taylor expansion series that can converge beyond 10 significant digits. It is needless to say that we need to calculate Taylor expansion constants only once and the option price would be an analytical expression constructed from a cumulative normal distribution function, an exponential function and finite sums. [ABSTRACT FROM PUBLISHER]

We introduce a general iterative algorithm for finding a common element of the common fixedpoint set of an infinite family of ?i-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan 2010 and many others. [ABSTRACT FROM AUTHOR]

Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper. [ABSTRACT FROM AUTHOR]

Recently, Xiao et al. proposed a nonsmooth equations-based method to solve the l₁-norm minimization problem (2011). The advantage of this method is its simplicity and lower storage. In this paper, based on new nonsmooth equations reformulation, we investigate new nonsmooth equations-based algorithms for solving l₁-norm minimization problems. Under mild conditions, we show that the proposed algorithms are globally convergent. The preliminary numerical results demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions. [ABSTRACT FROM AUTHOR]

The gradient projection algorithm plays an important role in solving constrained convex minimization problems. In general, the gradient projection algorithm has only weak convergence in infinite-dimensional Hilbert spaces. Recently, H. K. Xu (2011) provided two modified gradient projection algorithms which have strong convergence. Motivated by Xu's work, in the present paper, we suggest three more simpler variant gradient projection methods so that strong convergence is guaranteed. [ABSTRACT FROM AUTHOR]

Abstract: The main contribution of this paper is an algorithm to solve an extended version of the quantized consensus problem over networks represented by Hamiltonian graphs, i.e., graphs containing a Hamiltonian cycle, which we assume to be known in advance. Given a network of agents, we assume that a certain number of tokens should be assigned to the agents, so that the total number of tokens weighted by their sizes is the same for all the agents. The algorithm is proved to converge almost surely to a finite set containing the optimal solution. A worst case study of the expected convergence time is carried out, thus proving the efficiency of the algorithm with respect to other solutions recently presented in the literature. Moreover, the algorithm has a decentralized stop criterion once the convergence set is reached. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we study the identification of parametric Hammerstein systems with FIR linear parts. By a proper normalization and a clever characterization, it is shown that the average squared error cost function for identification can be expressed in terms of the inner product between the true but unknown parameter vector and its estimate. Further, the cost function is concave in the inner product and linear in the inner product square. Therefore, the identification of parametric Hammerstein systems with FIR linear parts is a globally convergent problem and has one and only one (local and global) minimum. This implies that the identification of such systems is a linear problem in terms of the inner product square and any local search based identification algorithm converges globally. [Copyright &y& Elsevier]

Abstract: This paper presents an augmented Lagrangian methodology with a stochastic population based algorithm for solving nonlinear constrained global optimization problems. The method approximately solves a sequence of simple bound global optimization subproblems using a fish swarm intelligent algorithm. A stochastic convergence analysis of the fish swarm iterative process is included. Numerical results with a benchmark set of problems are shown, including a comparison with other stochastic-type algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently. [Copyright &y& Elsevier]

Abstract: A comprehensive and systematic strategy for evaluating the performances of several trilinear second-order calibration algorithms is presented in this paper, in particular with a view of practical applications. Several trilinear second-order calibration methods such as PARAFAC, ATLD, SWATLD and APTLD, which have the “second-order advantage” and are gaining widespread acceptance in the field of chemometrics, were compared. Based on different input parameters including noise level, initial value, number of estimated components and collinearity in simulated and real data, the performances of these methods were evaluated in terms of predicting ability, consistency of resolved and real profiles, fitness obtained by selected components and speed of convergence. The obtained results give a reevaluation of the position and role of these trilinear second-order calibration methods in chemometrics and provide a guidance in practical applications for solving complicated quantitative analysis problems in analytical chemistry. It is useful and helpful to choose, for example, which algorithm would be more suitable for predicting the concentration of the analyte(s) of interest even in the presence of unknown interferents in complex systems. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an interval version of the generalized accelerated overrelaxation methods, which we refer to as IGAOR, for solving the linear complementarity problems, LCP (M, q), and develop a class of multisplitting IGAOR methods which can be easily implemented in parallel. In addition, in regards to the H-matrix with positive diagonal elements, we prove the convergence of these algorithms and illustrate their efficiency through our numerical results. [Copyright &y& Elsevier]

Abstract: Many real life problems can be stated as a minimax optimization problem, such as the problems in economics, finance, management, engineering and other fields. In this paper, we present an algorithm with nonmonotone strategy and second-order correction technique for minimax optimization problems. Using this scheme, the new algorithm can overcome the difficulties of the Maratos effect occurred in the nonsmooth optimization, and the global and superlinear convergence of the algorithm can be achieved accordingly. Numerical experiments indicate some advantages of this scheme. [Copyright &y& Elsevier]

Abstract: In this paper, we prove strong convergence theorems by the hybrid method for a family of hemi-relatively nonexpansive mappings in a Banach space. Our results improve and extend the corresponding results given by Qin et al. [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Haiyun Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi--nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051–1055], and at the same time, our iteration algorithm is different from the Kimura and Takahashi algorithm, which is a modified Mann-type iteration algorithm [Yasunori Kimura, Wataru Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl. 357 (2009) 356–363]. In addition, we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems. [ABSTRACT FROM AUTHOR]

The classical inexact Newton algorithm is all efficient and popular technique for solving large sparse nonlinear systems of equations. When the nonlinearities in the system are well balanced, a near quadratic convergence is often observed; however, if some of the equations are ranch more nonlinear than the others in the system, the convergence is much slower. The slow convergence (or sometimes divergence) is often determined by the small number of equations in the system with the highest nonlinearities. The idea of nonlinear preconditioning has been proven to be very useful. Through subspace nonlinear solves, the local high nonlinearities are removed, and the fast convergence can then be restored when the inexact Newton algorithm is called after the preconditioning. Recently a left preconditioned inexact Newton method was proposed in which the nonlinear function is replaced by a preconditioned function with more balanced nonlinearities. In this paper, we combine an inexact Newton method with a restricted additive Schwarz based nonlinear elimination. The new approach is easier to implement than the left preconditioned method since the nonlinear function does not have to be replaced, and, furthermore, the nonlinear elimination step does not have to be called at every outer Newton iteration. We show numerically that it perform well for, as an example, solving the incompressible Navier-Stokes equations with high Reynolds numbers and on machines with large numbers of processors. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study the use of alternating direction algorithms for several ℓ1-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis pursuit denoising problems of both unconstrained and constrained forms, and others. We present and investigate two classes of algorithms derived from either the primal or the dual form of ℓ1-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an ℓ1-problem into one having blockwise separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the augmented Lagrangian function of the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical experiments are performed, using randomized partial Walsh Hadamard sensing matrices, to demonstrate the versatility and effectiveness of the proposed approach. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points: (i) that algorithm speed should be evaluated relative to appropriate solution accuracy; and (ii) that when erroneous measurements possibly exist, the ℓ1-fidelity should generally be preferable to the ℓ2-fidelity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we suggest and analyze two projection methods (one implicit and one explicit) for finding a particular solution of a mixed equilibrium problem in a real Hilbert space. Furthermore, we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Utilizing the iterative algorithm, we obtain the strong convergence theorems for a finite family of nonexpansive mappings. Furthermore, the problem of image recovery is considered in the above result. Our results extend and improve the corresponding results. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of asymptotically -strict pseudo-contractions in the setting of real Hilbert spaces. By using our proposed scheme, we get a weak convergence theorem for a finite family of asymptotically -strict pseudo-contractions and then we modify these algorithm to have strong convergence theorem by using the two hybrid methods in the mathematical programming. Our results improve and extend the recent ones announced by Ceng, et al.’s result [L.C. Ceng, Al-Homidan, Q.H. Ansari and J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] Qin, Cho, Kang, and Shang, [X. Qin, Y. J. Cho, S. M. Kang, and M. Shang, A hybrid iterative scheme for asymptotically -strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009) 1902–1911] and other authors. [Copyright &y& Elsevier]

Monte Carlo methods are now widely used in solving various computational fluid dynamics systems. This paper presents an improved scrambled quasi-Monte Carlo method for solving fluid dynamics applications. In our parallel implementation we use an independent scrambled quasirandom sequence for each processor. We explore the use of both scrambled quasi-Monte Carlo and variance reduction methods to improve the accuracy for Monte Carlo schemes. We also present theoretical analyses and numerical experiments to validate our numerical algorithms. [ABSTRACT FROM AUTHOR]

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the convergence of the method. As an application we model an endoglacial conduit in which a phase change phenomenon takes place. [Copyright &y& Elsevier]

Many current Evolutionary Algorithms (EAs) suffer from a tendency to converge prematurely or stagnate without progress for complex problems. This may be due to the loss of or failure to discover certain valuable genetic material or the loss of the capability to discover new genetic material before convergence has limited the algorithm's ability to search widely. In this paper, the Hierarchical Fair Competition (HFC) model, including several variants, is proposed as a generic framework for sustainable evolutionary search by transforming the convergent nature of the current EA framework into a non-convergent search process. That is, the structure of HFC does not allow the convergence of the population to the vicinity of any set of optimal or locally optimal solutions. The sustainable search capability of HFC is achieved by ensuring a continuous supply and the incorporation of genetic material in a hierarchical manner, and by culturing and maintaining, but continually renewing, populations of individuals of intermediate fitness levels. HFC employs an assembly-line structure in which subpopulations are hierarchically organized into different fitness levels, reducing the selection pressure within each subpopulation while maintaining the global selection pressure to help ensure the exploitation of the good genetic material found. Three EAs based on the HFC principle are tested - two on the even-10-parity genetic programming benchmark problem and a real-world analog circuit synthesis problem, and another on the HIFF genetic algorithm (GA) benchmark problem. The significant gain in robustness, scalability and efficiency by HFC, with little additional computing effort, and its tolerance of small population sizes, demonstrates its effectiveness on these problems and shows promise of its potential for improving other existing EAs for difficult problems. A paradigm shift from that of most EAs is proposed: rather than trying to escape from local optima or delay convergence at a local optimum, HFC allows the emergence of new optima continually in a bottom-up manner, maintaining low local selection pressure at all fitness levels, while fostering exploitation of high-fitness individuals through promotion to higher levels. [ABSTRACT FROM AUTHOR]

Abstract: This paper proves the convergence of some generalized Schwarz algorithms for solvingthe obstacle problems with a T-monotone operator. Numerical results show that the generalized Schwarz algorithms converge faster than the classical Schwarz algorithms. [Copyright &y& Elsevier]

The Fermat location problem is to find a point whose sum of weighted distances from m given points (vertices) is a minimum. The best known method of solution is an iterative scheme devised by Weiszfeld in 1937, which converges to the unique minimum point unless one of the iterates happens to "land" on a nonoptimal vertex. The convergence proof of this scheme depends on two theorems, one of which (descent theorem) states that the objective function strictly decreases at each step. This paper extends the descent theorem by proving: (1) there is a "ball" whose radius and center depend on the Weiszfeld iteration, such that any algorithm whose iterates are "in the ball" or "on its surface" is a descent algorithm; (2) under certain circumstances, one or more of the vertices may be deleted, although the weight(s) are taken into account, and the Weiszfeld algorithm retains its descent property. In general there are several subsets of vertices which may be deleted, and for each subset, a corresponding iterate; (3) the convex hull of these iterates is such that all points within it have the descent property. Examples of the potential application of these extensions are given, including the construction of a modified Weiszfeld algorithm that without exception converges to the optimum. Beyond that, it is hoped the theorems may in time be useful in proving the descent property of yet to be discovered, very fast, nongradient algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the split equality problem (SEP) in Hilbert spaces. By converting it to a coupled fixed-point equation, we propose a new algorithm for solving the SEP. Whenever the convex sets involved are level sets of given convex functionals, we propose two new relaxed alternating algorithms for the SEP. The first relaxed algorithm is shown to be weakly convergent and the second strongly convergent. A new idea is introduced in order to prove strong convergence of the second relaxed algorithm, which gives an affirmative answer to Moudafi's question. Finally, preliminary numerical results show the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature. [ABSTRACT FROM AUTHOR]

Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, which brings much more difficulty. In this paper, based on the mid-point quadrature rule and composite two-point Gauss-Legendre quadrature rule, two finite difference schemes are established. Different from the previous works, which concerned only one- or two-dimensional problems with linear source terms, time-fractional wave equations of distributed-order whose source term is nonlinear in two and even three dimensions are considered. In addition, to improve the computational efficiency, the technique of alternating direction implicit (ADI) decomposition is also adopted. The unique solvability of the difference scheme is discussed, and the unconditional stability and convergence are analyzed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two- and three-dimensional cases. [ABSTRACT FROM AUTHOR]