Large-scale Nonlinear Equations Research Articles

A Projection Hestenes–Stiefel Method with Spectral Parameter for Nonlinear Monotone Equations and Signal Processing

A number of practical problems in science and engineering can be converted into a system of nonlinear equations and therefore, it is imperative to develop efficient methods for solving such equations. Due to their nice convergence properties and low storage requirements, conjugate gradient methods are considered among the most efficient for solving large-scale nonlinear equations. In this paper, a modified conjugate gradient method is proposed based on a projection technique and a suitable line search strategy. The proposed method is matrix-free and its sequence of search directions satisfies sufficient descent condition. Under the assumption that the underlying function is monotone and Lipschitzian continuous, the global convergence of the proposed method is established. The method is applied to solve some benchmark monotone nonlinear equations and also extended to solve ℓ 1 -norm regularized problems to reconstruct a sparse signal in compressive sensing. Numerical comparison with some existing methods shows that the proposed method is competitive, efficient and promising.

A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application

In this paper, we propose a derivative free algorithm for solving non-linear monotone equations with convex constraints. The proposed algorithm combines the method of spectral gradient and the projection method. We also modify the backtracking line search technique. The global convergence of the proposed method is guaranteed, under the mild conditions. Further, the numerical experiments show that the large-scale non-linear equations with convex constraints can be effectively solved with our method. The $L_{1}$ -norm regularized problems in signal reconstruction are studied by using our method.

A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems

Nonlinear systems present a quite complicate problem. As the number of dimensions increases, it becomes more difficult to find the solution of the problem. In this paper, a modified conjugate gradient method is designed that has a sufficient descent property and trust region property. It is interesting that the formula for search direction makes full use of the property of convex combination between the deepest descent algorithm and the classical LS conjugate gradient (CG) method. The global convergence property is established under certain conditions, and the numerical results show that the modified method is effective for normal nonlinear equations and image restoration problems.

A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations

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.

A norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations

In this paper, we propose a norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations without involving any information of the gradient or Jacobian matrix by using some approximate substitutions. The proposed algorithm is extended from an efficient three-term conjugate gradient method for solving unconstrained optimization problems, and inherits some nice properties such as simple structure, low storage requirements and symmetric property. Under some appropriate conditions, the global convergence is proved. Finally, the numerical experiments and comparisons show that the proposed algorithm is very effective for large-scale problems.

A family of conjugate gradient methods for large-scale nonlinear equations

In this paper, we present a family of conjugate gradient projection methods for solving large-scale nonlinear equations. At each iteration, it needs low storage and the subproblem can be easily solved. Compared with the existing solution methods for solving the problem, its global convergence is established without the restriction of the Lipschitz continuity on the underlying mapping. Preliminary numerical results are reported to show the efficiency of the proposed method.

Floquet analysis of spatially periodic thermocapillary convection in a low-Prandtl-number liquid bridge

Secondary instability of thermocapillary convection was investigated for a half-zone liquid bridge with low Prandtl number fluids. The liquid bridge was suspended between two cylindrical flat disks, which were maintained at different temperatures. The thermocapillary-driven flow formed an axisymmetric steady toroidal vortex. If the temperature difference between the two disks exceeded a certain threshold, the axisymmetric flow transitioned to an azimuthal periodic steady flow. The objective in the analysis of secondary instability is to examine the stability limit of this steady flow. Employing Floquet analysis, we obtained the neutral stability curves as a function of the aspect ratio (radius/height) of the liquid bridge and identified the critical modes. The critical modes of the Floquet analysis are referred to as critical Floquet modes and are classified by Floquet parameter β, which characterizes a set of azimuthal wave numbers of the Floquet mode. The Newton–Krylov method and Arnoldi method are implemented to solve the large-scale nonlinear equations and generalized eigenvalue problems, respectively. Two critical Floquet modes were observed having different Floquet parameters, β = 0 and β = 1, which appear preferential in the liquid bridge for high and low aspect ratios, respectively. The mode with β = 1 was steady for low aspect ratios and was able to change to an oscillatory mode with a frequency of oscillation that rapidly increased with increasing aspect ratio. We visualized the temperature and azimuthal velocity distributions of the critical Floquet modes and its Fourier components. We concluded that the mode with β = 1 leads to a pulsating oscillation, denoted as “P-type,” whereas that with β = 0 leads to a twisting oscillation, denoted as “T-type.” Both types of oscillations were reported in previous studies. The present neutral stability curves are for the most part in good agreement with critical Reynolds numbers previously obtained in numerical simulations.

Fast and reliable computational strategy for developing a rigorous model-driven soft sensor of dynamic molecular weight distribution

Key polymer properties are substantially directly related to the polymer molecular weight distribution (MWD). On-line monitoring and prediction of dynamic MWD profiles are highly important for on-line quality control of polymerization processes. In this study, a fast and reliable computational strategy for an equation-oriented model-based soft sensor for the high-density polyethylene grade transition process is developed. The simultaneous collocation approach is adopted to discretize the dynamic model. A novel moving finite element method is proposed to improve the on-line performance of the derived large-scale nonlinear equation systems. The sensitivity information of the nonlinear equation systems contributes to a convergence enhancement strategy for the sensor. The prediction accuracy and computational efficiency are demonstrated using industrial data. A potential application to extend the polymerization process with changeable flowsheet is also tested through simulation.

A modified conjugate gradient algorithm with backtracking line search technique for large-scale nonlinear equations

ABSTRACTConjugate gradient methods are widely used for solving unconstrained optimization and nonlinear equations, specially in large-scale cases. Since they own the attractive practical factors of simple computation and low memory requirement, interesting theoretical features of curvature information and strong global convergence. In this paper, we present a modified conjugate gradient algorithm by line search method with acceleration scheme for nonlinear symmetric equations. Furthermore, the proposed method not only possess descent property but also owns global convergence in mild conditions. Numerical results also indicate that the presented method is much more effective than the other methods for the test problems.

A quasi-Newton algorithm for large-scale nonlinear equations

In this paper, the algorithm for large-scale nonlinear equations is designed by the following steps: (i) a conjugate gradient (CG) algorithm is designed as a sub-algorithm to obtain the initial points of the main algorithm, where the sub-algorithm’s initial point does not have any restrictions; (ii) a quasi-Newton algorithm with the initial points given by sub-algorithm is defined as main algorithm, where a new nonmonotone line search technique is presented to get the step length alpha_{k}. The given nonmonotone line search technique can avoid computing the Jacobian matrix. The global convergence and the 1+q-order convergent rate of the main algorithm are established under suitable conditions. Numerical results show that the proposed method is competitive with a similar method for large-scale problems.

Global Convergence Analysis of a Nonlinear Conjugate Gradient Method for Unconstrained Optimization Problems

Background/Objectives: The Conjugate Gradient (CG) methods are the well-known iterative methods use for finding solutions to nonlinear system equations. There is need to address the jamming phenomenal facing the current class of this methods. Methods/Statistical Analysis: In order to address the shortcomings, we work on the denominator of the Yao et al., CG method which is known to generate descent direction for objective functions by proposing an entire different CG coefficient which can easily switch in case jamming occurs by imposing some parameters thereby guarantee global convergence. Findings: The proposed CG formula performs better than classical methods as well as Yao et al. Under Wolfe line search condition, the convergence analysis of the proposed CG formula was established. Some benchmark problems from cute collections are used as basis of strength comparisons of the proposed formula against some other CG formulas. Effectiveness and efficiency of the obtained results for the proposed formula is clearly shown by adopting the performance profile of Dolan and More’ which is one of most acceptable techniques of strength comparisons among methods. Application: Mathematicians and Engineers who are interested in finding solutions to large scale nonlinear equations can apply the method leading to global optimization dealing with best possible solutions ever for given problems.

A new nonmonotone spectral residual method for nonsmooth nonlinear equations

In this paper, a new spectral residual method is proposed to solve systems of large-scale nonlinear equations, where the steplength is obtained by minimizing the residue of an approximate secant equation. Especially, the new steplength can be directly applied into solving strictly convex quadratic function. Combined with a new nonmonotone line search strategy, a new derivative-free algorithm, called a nonmonotone spectral residual algorithm (NSRA), is developed. Under mild assumptions, global convergence is established for locally Lipschitz continuous nonlinear systems. Compared with the state-of-the-art algorithms available in the literatures, the new algorithm is more efficient in solving large-scale benchmark test problems.

A trust region spectral method for large-scale systems of nonlinear equations

The spectral gradient method is one of the most effective methods for solving large-scale systems of nonlinear equations. In this paper, we propose a new trust region spectral method without gradient. The trust region technique is a globalization strategy in our method. The global convergence of the proposed algorithm is proved. The numerical results show that our new method is more competitive than the spectral method of La Cruz et al. (Math. Comput. 75(255):1429-1448, 2006) for large-scale nonlinear equations.

Two New PRP Conjugate Gradient Algorithms for Minimization Optimization Models.

Two new PRP conjugate Algorithms are proposed in this paper based on two modified PRP conjugate gradient methods: the first algorithm is proposed for solving unconstrained optimization problems, and the second algorithm is proposed for solving nonlinear equations. The first method contains two aspects of information: function value and gradient value. The two methods both possess some good properties, as follows: 1)β k ≥ 0 2) the search direction has the trust region property without the use of any line search method 3) the search direction has sufficient descent property without the use of any line search method. Under some suitable conditions, we establish the global convergence of the two algorithms. We conduct numerical experiments to evaluate our algorithms. The numerical results indicate that the first algorithm is effective and competitive for solving unconstrained optimization problems and that the second algorithm is effective for solving large-scale nonlinear equations.

Incomplete Newton-Ulm method for large scale nonlinear equations

This paper presents an incomplete Newton-Ulm method (INU) for nonlinear equations. This method uses parts of elements of Jacobian matrix to obtain the next iteration point and does not contain inverse operators in its expression. We discuss and analyze the convergence conditions and semilocal convergence of the new method. Some special INU algorithms are designed and numerical experiments are given. Numerical results show that INU method is effective for large scale nonlinear equations.

A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations

It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes---Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes---Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

A Limited-Memory BFGS Algorithm Based on a Trust-Region Quadratic Model for Large-Scale Nonlinear Equations.

In this paper, a trust-region algorithm is proposed for large-scale nonlinear equations, where the limited-memory BFGS (L-M-BFGS) update matrix is used in the trust-region subproblem to improve the effectiveness of the algorithm for large-scale problems. The global convergence of the presented method is established under suitable conditions. The numerical results of the test problems show that the method is competitive with the norm method.

On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure

For the large-scale nonlinear matrix equations with low rank structure, the well-developed doubling algorithm in low rank form (DA-LR) is known as an efficient method to compute the stabilizing solution. By further analyzing the global efficiency index constructed in this paper, we propose a tripling algorithm in low rank form (TA-LR) from two points of view, the cyclic reduction and the symplectic structure preservation. The new presented algorithm shares the same pre-processing complexity with that of DA-LR, but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset. Under the solvability condition, the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order. Numerical experiments including some from nano research show that the TA-LR is highly efficient to compute the stabilizing solution of large-scale nonlinear matrix equations with low rank structure.

A three-terms Polak–Ribière–Polyak conjugate gradient algorithm for large-scale nonlinear equations

In this paper, a conjugate gradient algorithm for systems of large-scale nonlinear equations is designed by the following steps: (i) A three-terms conjugate gradient direction dk is presented where the direction possesses the sufficient descent property and the trust region property independent of line search technique; (ii) A backtracking line search technique along the direction is proposed to get the step length αk and construct a point; (iii) If the point satisfies the given condition then it is the next point, otherwise the projection-proximal technique is used and get the next point. Both the direction and the line search technique are the derivative-free approaches, then the large-scale nonlinear equations are successfully solved (100,000 variables). The global convergence of the given algorithm is established under suitable conditions. Numerical results show that the proposed method is efficient for large-scale problems.

Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints

In this paper, the Wei–Yao–Liu (WYL) conjugate gradient projection algorithm will be studied for nonlinear monotone equations with convex constraints, which can be viewed as an extension of the WYL conjugate gradient method for solving unconstrained optimization problems. These methods can be applied to solving large-scale nonlinear equations due to the low storage requirement. We can obtain global convergence of our algorithm without requiring differentiability in the case that the equation is Lipschitz continuous. The numerical results show that the new algorithm is efficient.