Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation

We propose a tensor structured preconditioner for the tensor train GMRES algorithm (or TT-GMRES for short) to approximate the solution of the all-at-once formulation of time-dependent fractional partial differential equations discretized in time by linear multistep formulas used in boundary value form and in space by finite volumes.Numerical experiments show that the proposed preconditioner is efficient for very large problems and is competitive, in particular with respect to the AMEn algorithm.     

Tensor complementarity problems: the GUS-property and an algorithm

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.     

A generalized Newton method for absolute value equations associated with second order cones

In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.     

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

This article is concerned with solving the high order Stein tensor equation arising in control theory. The conjugate gradient squared (CGS) method and the biconjugate gradient stabilized (BiCGSTAB) method are attractive methods for solving linear systems. Compared with the large-scale matrix equation, the equivalent tensor equation needs less storage space and computational costs. Therefore, we present the tensor formats of CGS and BiCGSTAB methods for solving high order Stein tensor equations. Moreover, a nearest Kronecker product preconditioner is given and the preconditioned tensor format methods are studied. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.     

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane–Emden equations

The purpose of this paper is to present a numerical algorithm for solving the Lane–Emden equations as singular initial value problems. The proposed algorithm is based on an operational Tau method (OTM). The main idea behind the OTM is to convert the desired problem to some operational matrices. Firstly, we use a special integral operator and convert the Lane–Emden equations to integral equations. Then, we use OTM to linearize the integral equations to some operational matrices and convert the problem to an algebraic system. The concepts, properties, and advantages of OTM and its application for solving Lane–Emden equations are presented. Some orthogonal polynomials are also used to reduce the volume of computations. Finally, several experiments of Lane–Emden equations including linear and nonlinear terms are given to illustrate the validity and efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

A new smoothing Newton method for solving constrained nonlinear equations

In this paper, a new smoothing Newton method is proposed for solving constrained nonlinear equations. We first transform the constrained nonlinear equations to a system of semismooth equations by using the so-called absolute value function of the slack variables, and then present a new smoothing Newton method for solving the semismooth equations by constructing a new smoothing approximation function. This new method is globally and quadratically convergent. It needs to solve only one system of unconstrained equations and to perform one line search at each iteration. Numerical results show that the new algorithm works quite well.     

Numerical algorithm for solving diffusion-type equations on the basis of multigrid methods

A new numerical algorithm based on multigrid methods is proposed for solving equations of the parabolic type. Theoretical error estimates are obtained for the algorithm as applied to a two-dimensional initial-boundary value model problem for the heat equation. The good accuracy of the algorithm is demonstrated using model problems including ones with discontinuous coefficients. As applied to initial-boundary value problems for diffusion equations, the algorithm yields considerable savings in computational work compared to implicit schemes on fine grids or explicit schemes with a small time step on fine grids. A parallelization scheme is given for the algorithm.     

Generalized cauchy singular integral for the boundary values of two-dimensional flows in an anisotropic-inhomogeneous layer of a porous medium

We consider the main boundary value problems of two-dimensional stationary flows in an anisotropic-inhomogeneous layer with an arbitrary (not necessarily symmetric) permeability tensor. We present Cauchy integrals and Cauchy type integrals whose kernels can be expressed via the fundamental solutions of the main equations and have a hydrodynamic meaning. This permits one to develop the method of singular integral equations for solving two-dimensional boundary value problems. The considered problems can be used as mathematical models, in particular, for the extraction of fluids (water, oil) from natural layers of soil with complicated geological structure.     

Singular integrals with generalized Cauchy kernels for the velocity field in boundary value problems of filtration in an anisotropic inhomogeneous porous layer

We study two-dimensional stationary and nonstationary boundary value problems of fluid filtration in an anisotropic inhomogeneous porous layer whose conductivity is modeled by a not necessarily symmetric tensor. For the velocity field, we introduce generalized singular Cauchy and Cauchy type integrals whose kernels are expressed via the leading solutions of the main equations and have a hydrodynamic interpretation. We obtain the limit values of a Cauchy type generalized integral (Sokhotskii-Plemelj generalized formulas). This permits one to develop a method for solving boundary value problems for the filtration velocity field. The idea of the method and its efficiency are illustrated for the boundary value problem of filtration in adjacent layers of distinct conductivities and the problem of the evolution of liquid interface.     

Solving nonlinear equations with the convex combination of two positive spectral coefficients

We present a spectral algorithm based on the convex combination of two modified spectral coefficients for solving systems of nonlinear equations. The proposed algorithm does not require the exact or approximated directional derivative for its implementation. By employing a derivative-free line search, the global convergence of the sequence generated by the algorithm is supported. Numerical experiments are given to demonstrate the performance of the algorithm compared with a similar algorithm in the literature for solving nonlinear equations problems.     

Formulating an <Emphasis Type="Italic">n</Emphasis>-person noncooperative game as a tensor complementarity problem

In this paper, we consider a class of n-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the multilinear game is equivalent to finding a solution of the resulted tensor complementarity problem. Especially, we present an explicit relationship between the solutions of the multilinear game and the tensor complementarity problem, which builds a bridge between these two classes of problems. We also apply a smoothing-type algorithm to solve the resulted tensor complementarity problem and give some preliminary numerical results for solving the multilinear games.     

一类显式的k阶线性k步法基本公式

本文给出了一类比Adams-Bashforth方法的局部截断误差主项系数小和绝对稳定区间大的显式k阶线性k步法基本公式.作者求出了公式的分数形式的精确系数,阶数和局部截断误差主项系数,给出了3-9步公式的绝对稳定区间,构造了由新公式的4阶显式公式和一个同阶隐式基本公式组合而成的特殊预估-校正方法,它的绝对稳定区间大于预估公式而且等于校正公式, 比著名的Adams-Bashforth-Moulton预估校正方法的绝对稳定区间大, 最后用数值试验对结果进行了验证,适合于求解常微分方程初值问题.     

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.     

求解非线性单调方程组的一种无导数投影算法

推广了一种修正的CG_DESCENT共轭梯度方法,并建立了一种有效求解非线性单调方程组问题的无导数投影算法.在适当的线搜索条件下,证明了算法的全局收敛性.由于新算法不需要借助任何导数信息,故它适应于求解大规模非光滑的非线性单调方程组问题.大量的数值试验表明,新算法对给定的测试问题是有效的.     

A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations

In this paper, a modified nonmonotone BFGS algorithm is developed for solving a smooth system of nonlinear equations. Different from the existent techniques of nonmonotone line search, the value of an algorithmic parameter controlling the magnitude of nonmonotonicity is updated at each iteration by the known information of the system of nonlinear equations such that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence of the algorithm is established for solving a generic nonlinear system of equations. Implementing the algorithm to solve some benchmark test problems, the obtained numerical results demonstrate that it is more effective than some similar algorithms available in the literature.     

Numerical algorithm for solving diffusion equations on the basis of multigrid methods

A new effective algorithm based on multigrid methods is proposed for solving parabolic equations. The algorithm preserves implicit-scheme advantages (such as stability, accuracy, and conservativeness) while it involves a considerably reduced amount of arithmetic operations at every time level. The absolute stability, conservativeness, and convergence of the algorithm is proved theoretically using one- and two-dimensional initial-boundary value model problems for the heat equation. The error of the solution is estimated. The good accuracy of the method is demonstrated using two-dimensional model problems, including ones with discontinuous coefficients.     

Absolute value programming

We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax+B|x| = b, where A and B are arbitrary m× n real matrices. We show that this absolute value equation is NP-hard to solve, and that solving it with B = I solves the general linear complementarity problem. We give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities. We also propose concave minimization formulations for absolute value equations that are solved by a finite succession of linear programs. These algorithms terminate at a local minimum that solves the absolute value equation in almost all solvable random problems tried.     

Tensor-Krylov methods for large nonlinear equations

In this paper, we describe tensor methods for large systems of nonlinear equations based on Krylov subspace techniques for approximately solving the linear systems that are required in each tensor iteration. We refer to a method in this class as a tensor-Krylov algorithm. We describe comparative testing for a tensor-Krylov implementation versus an analogous implementation based on a Newton-Krylov method. The test results show that tensor-Krylov methods are much more efficient and robust than Newton-Krylov methods on hard nonlinear equations problems.Part of this work was performed while the author was research associate at CERFACS (Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique).Research supported in part by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Decomposition of linear-quadratic optimal control problems for two-steps systems

The paper deals with problems with state equations of two sequentially acting systems. Matching conditions for trajectories at the switch point are absent, however the minimized functional depends on values of a state trajectory in the left and right sides from the switch point. State trajectories have fixed left and right points, and, in general, they are discontinuous functions. The algorithm for solving this problem is given, which is based on sequential solving eight independent initial value problems. The formula for the minimal value of the performance index is also presented.