Sharif Digital Repository

The main goal of this work is solving system of linear equations Ax = b, where A is a n_n square matrix, b is a n_1 vector and x is the vector of unknowns. When n is large, using direct methods is not economical. Thus, the system is solved by iterative methods. At first, projection method onto subspace K _ Rn with dimension m _ n is described, and then this subspace K is equalized with the krylov subspace. Then,some samples of projection methods onto the krylov subspace, such as FOM, GMRES and CG (Conjugate Gradient), are considered. The preconditioning of the linear system is explained, that is, instead of solving system Ax = b, the system PAx = Pb (P nonsingular), is solved, such that the... 

Time-dependent origin-destination demand is a key input in dynamic traffic assignment in advanced traffic management systems, and the result of dynamic traffic assignment is dependent on the accuracy of this information. One method to achieve time dependent demand matrices is using a primary demand matrix and volume traffic counts in some links of network. In this thesis a bi-level model is used to correct the demand matrix and the extended gradient method is suggested to solve the problem. The extended gradient is an iterative method that in each iteration, corrects the demand matrix in a way that the estimated traffic flow be close to the observed traffic flow. Execution of this method in... 

Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of matrices. Recently, a new conjugacy condition proposed by Dai and Liao, considers an inexact line search scheme that reduces to the old one if the line search is exact. Based on this condition, a new conjugate gradient method was proposed that has fast convergence. Later, Yabe and Takano, based on new conjugacy condition and modified secant condition, proposed another conjugate gradient method. This method takes both the available gradient and function value information and achieves a high-order accuracy in approximating the second-order curvature of the objective... 

We discuss conjugate gradient methods for which both the gradient and func-tion values are considered in computing the conjugate gradient parameter. We pro-pose new conjugate gradient methods as members of Dai-Liao’s family of conjugate gradient methods and Andrei’s family of hybrid conjugate gradient methods. For computing the conjugate gradient parameter in our methods, three modified secant equations proposed by Zhang, Deng and Chen, Li and Fukushima, and Yuan are used. It is shown that under proper conditions, three of the proposed methods are globally convergent for uniformly convex functions and two other methods are glob-ally convergent for general functions. It is also shown that... 

Conjugate gradient methods are useful for large scale nonlinear optimization problem, because they avoid the storage of any matrices. In this thesis, we have investigated an accelerated hybrid conjugate gradient algorithm, recently proposed in the literature. The combining parameter is calculated so that the corresponding direction to the conjugate gradient algorithm, while satisfies the modified secant condition, is a Newton direction. It is shown that for uniformly convex functions and for general nonlinear functions the algorithm with strong Wolfe line search is globally convergent. The algorithm uses an accelerated approach for the reduction of the objective function values by modifying... 

An interior point method for nonlinear programming problem recently proposed by Waltz, Morales, Nocedal and Orban is described and implemented [2].The steps are computed by line search based on the primal-dual equations, or trust region based on the conjugate gradient iteration. Steps computed by line search are tried first, but if they are determinded to be ineffective, a trust region iteration that guarantees progress toward a stationary point is used. In order to reduce the calculations, here we propose some modifications. The algorithms are implemented and the programs are tested on a variety of problems. Numerical results based on Dolan-More’ confirm the effectiveness of the algorithms  

Numerical Simulation of two-dimensional steady compressible fluid flow on unstructured grids was accomplished using a fast implicit algorithm. To solve the copmlete two-dimensional Navier-Stokes equations, implicit time stepping was used which results in a large sparse linear system in each iteration. To solve the linear system, the biconjugate gradient method which belongs to Krylov subspace methods family, with an ILU(0) preconditioner was used. For accelerating the solution in large problems, parallel processing was used for linear system to be solved faster. Two upwind methods, namely Roe’s and AUSM+ methods were used for spatial descritizaion of inviscid fluxes with a MUSCL algorithm... 

We describe two modified secant equations proposed by Yuan, Li and Fukushima. First, we study the approach proposed by Andrei. Then, we explain two hybrid conjugate gradient methods for unconstrained optimization problems. The methods are hybridizations of Hestenes-Stiefel and Dai-Yuan conjugate gradient methods. It is shown that one of the algorithms is globally convergent for uniformly convex functions and the other is globally convergent for general functions. Two approaches for computing the initial value of the steplength proposed by Babaie, Fatemi, and Mahdavi-Amiri and Andrei are used for accelerating the performance of the line search. We implement the algorithms and compare the...