The use of associative information processing for constructing algorithms for solving systems of Boolean equations

Possibilities of the application of dedicated processors based on the use of associative memory for solving systems of Boolean equations is considered. An algorithm for solving systems of Boolean equations using associative dedicated processors is proposed. Classes of systems of Boolean equations that can be efficiently solved by this algorithm are found. Subexponential estimates of the expectation of the computational complexity of the proposed algorithm for solving systems of equations belonging to these classes are obtained.     

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.     

A constrained optimization approach to solving certain systems of convex equations

This research presents a new constrained optimization approach for solving systems of nonlinear equations. Particular advantages are realized when all of the equations are convex. For example, a global algorithm for finding the zero of a convex real-valued function of one variable is developed. If the algorithm terminates finitely, then either the algorithm has computed a zero or determined that none exists; if an infinite sequence is generated, either that sequence converges to a zero or again no zero exists. For solving n-dimensional convex equations, the constrained optimization algorithm has the capability of determining that the system of equations has no solution. Global convergence of the algorithm is established under weaker conditions than previously known and, in this case, the algorithm reduces to Newton’s method together with a constrained line search at each iteration. It is also shown how this approach has led to a new algorithm for solving the linear complementarity problem.     

An two phase abs method for solving over determined systems of linear inequalities

The method, called the Multi-Stage ABS algorithm, for solving the over-determined linear inequalities system and the system combined with the over-determined linear inequalities and the equations is presented. This method is characterized by translating inequalities system to an equations system with slack variables. The explicit solution with the slack variables of the equations system are given by the implicit LU algorithm, then the slack variables can be given by the ABS algorithm. Finally, the upper multiplications of the algorithm are given.     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

An EM Algorithm for Estimating Equations

This article presents an algorithm for accommodating missing data in situations where a natural set of estimating equations exists for the complete data setting. The complete data estimating equations can correspond to the score functions from a standard, partial, or quasi-likelihood, or they can be generalized estimating equations (GEEs). In analogy to the EM, which is a special case, the method is called the ES algorithm, because it iterates between an E-Step wherein functions of the complete data are replaced by their expected values, and an S-Step where these expected values are substituted into the complete-data estimating equation, which is then solved. Convergence properties of the algorithm are established by appealing to general theory for iterative solutions to nonlinear equations. In particular, the ES algorithm (and indeed the EM) are shown to correspond to examples of nonlinear Gauss-Seidel algorithms. An added advantage of the approach is that it yields a computationally simple method for estimating the variance of the resulting parameter estimates.     

AN EFFECT ITERATION ALGORITHM FOR NUMERICAL SOLUTION OF DISCRETE HAMILTON-JACOBIBELLMAN EQUATIONS

§1Introduction ConsidertheHamilton-Jacobi-Bellmanequation max1≤v≤m[A(v)u(x)-f(v)(x)]=0,x∈Ω(1.1)withtheboundarycondition u(x)=0,x∈Ω(1.2)whereΩisabounded,smoothdomaininEuclideanspaceRd,d∈N;f(v)(x)aregiven functionsfromC2(Ω);A(v)aresecond-orderuniformlyellipticoperatorsoftheform A(v)=-d i,j=1a(v)ij2xixj+di=1b(v)ixi+c(v).(1.3)Intheaboveexpression(1.3)therearecoefficientsa(v)ij,b(v)i,c(v)∈C2(Ω)satisfying,forall1≤v≤m,a(v)ij(x)=a(v)ji(x),1≤i,j≤d,c(v)≥c0≥0,x∈Ω,a…     

A pod reduced-order spdmfe extrapolating algorithm for hyperbolic equations

In this article, a proper orthogonal decomposition (POD) method is used to study a classical splitting positive definite mixed finite element (SPDMFE) formulation for second-order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.     

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 Algorithm for Non-linear Simultaneous Equations

An algorithm for solving just-determined or underdeterminednon-linear simultaneous equations is presented. The algorithm,which is a kind of continuation method, consists of findingiteratively the minimum norm correction of the parameter-imbeddedlinearized equations. Its automatic scaling property frees usersfrom scaling the equations and unknowns. The notion of non-linearityweights has been introduced in connection with the non-linearityof each equation, and with deciding whether or not an obtainedsolution is promising. The imbedded parameters are also controlledby the non-linearity. The algorithm is very effective for suchcases as Rosenbrock's parabolic valley, where both hnear andnon-linear equations are included in the system.     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

Differential Elimination–Completion Algorithms for DAE and PDAE

Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics.
A DAE has differential index r if a minimum of r +1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties.
Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.     

一类特征值反问题(IEP)的基于矩阵方程的Ulm型算法

应用求解算子方程的Ulm方法构造了求解一类矩阵特征值反问题（IEP）的新算法.所给算法避免了文献[Aishima K.,A quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems,Linear Algebra and its Applications,2018,542：310-33]中算法在每次迭代中要求解一个线性方程组的不足,证明了在给定谱数据互不相同的条件下所给算法具有根收敛意义下的二次收敛性.数值实验表明本文所给算法在矩阵阶数较大时计算效果优于上文所给算法.     

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.     

A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR HYPERBOLIC EQUATIONS

In this article, a proper orthogonal decomposition （POD） method is used to study a classical splitting positive definite mixed finite element （SPDMFE） formulation for second- order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.     

求解特征值互补问题的一类ABS算法

ABS算法是20世纪80年代初,由Abaffy,Broyden和Spedicato完成的用于求解线性方程组的含有三个参量的投影算法,是一类有限次迭代直接法。目前,ABS算法不仅可以求解线性与非线性方程组,还可以求解线性规划和具有线性约束的非线性规划等问题。本文即是利用ABS算法求解特征值互补问题的一种尝试,构造了求解特征值互补问题的ABS算法,证明了求解特征值互补问题的ABS算法的收敛性。数值例子充分验证了求解特征值互补问题的ABS算法的有效性。     

Inexactly constrained discrete adjoint approach for steepest descent-based optimization algorithms

The problem of constrained optimization via the gradient-based discrete adjoint steepest descent method is studied under the assumption that the constraint equations are solved inexactly. Error propagation from the constraint equations to the gradient is studied analytically, as is the convergence rate of the inexactly constrained algorithm as it relates to the exact algorithm. A method is developed for adapting the residual tolerance to which the constraint equations are solved. The adaptive tolerance method is applied to two simple test cases to demonstrate the potential gains in computational efficiency.     

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.     

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.     

不等式约束优化一个可行序列线性方程组算法

提出了求解非线性不等式约束优化问题的一个可行序列线性方程组算法. 在每次迭代中, 可行下降方向通过求解两个线性方程组产生, 系数矩阵具有较好的稀疏性. 在较为温和的条件下, 算法具有全局收敛性和强收敛性, 数值试验表明算法是有效的.