Nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems

In this paper, we deal with conjugate gradient methods for solving nonlinear least squares problems. Several Newton-like methods have been studied for solving nonlinear least squares problems, which include the Gauss-Newton method, the Levenberg-Marquardt method and the structured quasi-Newton methods. On the other hand, conjugate gradient methods are appealing for general large-scale nonlinear optimization problems. By combining the structured secant condition and the idea of Dai and Liao (2001) [20], the present paper proposes conjugate gradient methods that make use of the structure of the Hessian of the objective function of nonlinear least squares problems. The proposed methods are shown to be globally convergent under some assumptions. Finally, some numerical results are given.     

Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

A modified conjugate gradient algorithm with cyclic Barzilai–Borwein steplength for unconstrained optimization

For solving large-scale unconstrained minimization problems, the nonlinear conjugate gradient method is welcome due to its simplicity, low storage, efficiency and nice convergence properties. Among all the methods in the framework, the conjugate gradient descent algorithm — CG_DESCENT is very popular, in which the generated directions descend automatically, and this nice property is independent of any line search used. In this paper, we generalize CG_DESCENT with two Barzilai–Borwein steplength reused cyclically. We show that the resulting algorithm owns attractive sufficient descent property and converges globally under some mild conditions. We test the proposed algorithm by using a large set of unconstrained problems with high dimensions in CUTEr library. The numerical comparisons with the state-of-the-art algorithm CG_DESCENT illustrate that the proposed method is effective, competitive, and promising.     

PRECONDITIONING HIGHER ORDER FINITE ELEMENT SYSTEMS BY ALGEBRAIC MULTIGRID METHOD OF LINEAR ELEMENTS

We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.     

Conjugate points at infinity in infinite dimensional teichmüller spaces

We show in this article that there exists conjugate points at infinity in infinite dimensional Teichmüller spaces.     

SINE TRANSFORM MATRIX FOR SOLVING TOEPLITZ MATRIX PROBLEMS

1. IntroductionStrang[1] first studied the use of circulallt matrices C for solving systems of linear eqllationsTi x = b witha symmetric positive definite Toeplitz matrix.Numerous authors such as T.Chan[2],R.Chan,etc.[3],[4],[5], Tyrtyshnikov[6], Huckle[7] and T.Ku and C.Kuo[8] proposed differentfamilies of circulallt / skew- circulant precondit ioners.Appling the preconditioned conjugate gradient algorithm(PCGA) to solve the systems Ti x -b, we must find a preconditioner P such that P…     

A sufficient descent LS conjugate gradient method for unconstrained optimization problems

In this paper, we make a modification to the Liu-Storey (LS) conjugate gradient method and propose a descent LS method. The method can generate sufficient descent directions for the objective function. This property is independent of the line search used. We prove that the modified LS method is globally convergent with the strong Wolfe line search. The numerical results show that the proposed descent LS method is efficient for the unconstrained problems in the CUTEr library.     

自反Banach空间中的相补问题

在自反Banach空间中证明了相补问题解的存在性定理，改进了[1]中的一个主要结果．     

Optimal shape design for packaging containing heating elements by inverse heat transfer method

The present study is concerned with the determination of the optimal shape for a package containing multiple heating elements. The optimization tool has been developed based on the inverse heat transfer (IHT) approach, incorporating a direct problem solver, a numerical grid generator, a direct-differentiation sensitivity analyzer, and the conjugate gradient method. Shape design that leads to a specified outer surface temperature distribution is predicted by the approach. In this study, the effects of internal heat generation on optimal shapes of the packagings have also been evaluated. Several practical cases with various imbedded heating elements and thermal conditions are studied. Results show that the approach provides an efficient computer-aided design scheme for the shape profile determination.     

A Trust Region Algorithm with Conjugate Gradient Technique for Optimization Problems

By means of a conjugate gradient strategy, we propose a trust region method for unconstrained optimization problems. The search direction is an adequate combination of the conjugate gradient direction and the trust-region direction. The global convergence and the quadratic convergence of this method are established under suitable conditions. Numerical results show that the presented method is competitive to the trust region method and the conjugate gradient method.