Search Results (4)

GMRES is one of the most powerful and popular methods to solve linear systems in the Krylov subspace; we examine it from two viewpoints: to maximize the decreasing length of the residual vector, and to maintain the orthogonality of the consecutive residual vector. A stabilization factor, $\eta$, to measure the deviation from the orthogonality of the residual vector is inserted into GMRES to preserve the orthogonality automatically. The re-orthogonalized GMRES (ROGMRES) method guarantees the absolute convergence; even the orthogonality is lost gradually in the GMRES iteration. When $\eta <1/2$, the residuals’ lengths of GMRES and GMRES(m) no longer decrease; hence, $\eta <1/2$ can be adopted as a stopping criterion to terminate the iterations. We prove $\eta =1$ for the ROGMRES method; it automatically keeps the orthogonality, and maintains the maximality for reducing the length of the residual vector. We improve GMRES by seeking the descent vector to minimize the residual in a larger space of the affine Krylov subspace. The resulting orthogonalized maximal projection algorithm (OMPA) is identified as having good performance. We further derive the iterative formulas by extending the GMRES method to the affine Krylov subspace; these equations are slightly different from the equations derived by Saad and Schultz (1986). The affine GMRES method is combined with the orthogonalization technique to generate a powerful affine GMRES (A-GMRES) method with high performance. Full article

Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media. In addition, to solve systems of linear equations in such problems, the generalized minimal residual method (GMRES) is often used. This paper analyzed the one-dimensional problem of multicomponent fluid flow in a porous medium and solved the system of the algebraic equation with the Newton-GMRES method. We calculated the linear equations with the GMRES, the GMRES with restarts after every m steps—GMRES (m) and preconditioned with Incomplete Lower-Upper factorization, where the factors L and U have the same sparsity pattern as the original matrix—the ILU(0)-GMRES algorithms, respectively, and compared the computation time and convergence. In the course of the research, the influence of the preconditioner and restarts of the GMRES (m) algorithm on the computation time was revealed; in particular, they were able to speed up the program. Full article

When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential $\varphi$ depends on the electrostatic charge $q_e$, as expressed by the equation $\mathcal{L}\left(q_e\right)=\varphi$, where $\mathcal{L}$ is an integral operator. Method of moments (MoM) is an excellent candidate for solving $\mathcal{L}\left(q_e\right)=\varphi$ numerically. In fact, considering $q_e$ as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, $q_e={\sum }_{n=1}^N{\alpha }_n{f}_n$ (with weights ${\alpha }_n$ and N, number of the subsections of the conductor) defined in the $\mathcal{L}$ domain so that $\varphi$ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system $L_{mn}{\alpha }_n=g_m$ with dense matrix, equivalent to $\mathcal{L}\left(q_e\right)=\varphi$. Once solved, the linear algebraic system gets ${\alpha }_n$ and therefore $q_e$ is obtainable so that the electrostatic capacitance $C=q_e/V$, where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve $L_{mn}{\alpha }_n=g_m$. These methods have, as a basic idea, the projection of a problem related to a matrix $A\in {\mathbb{R}}^{n\times n}$, having a number of non-null elements of the order of n, in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine $L_{mn}$ according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function; (2) pulse function as the basis function as well as testing function; (3) triangular function as the basis function and delta function as the testing function; (4) triangular function as the basis function and pulse function as the testing function; (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five $L_{mn}$ and five pair $q_e$ and C were computed. For each case, for the resolution of $L_{mn}{\alpha }_n=g_m$ obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab® Toolbox to evaluate $q_e$ and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of $L_{mn}$ and considerations on C, to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss–Seidel approaches. Full article

To solve large scale linear equations involved in the Fast Multipole Boundary Element Method (FM-BEM) efficiently, an iterative method named the generalized minimal residual method (GMRES(m)) algorithm with Variable Restart Parameter (VRP-GMRES(m)) algorithm is proposed. By properly changing a variable restart parameter for the GMRES(m) algorithm, the iteration stagnation problem resulting from improper selection of the parameter is resolved efficiently. Based on the framework of the VRP-GMRES(m) algorithm and the relevant properties of generalized inverse matrix, the projection of the error vector $r_{m+1}$ on $r_m$ is deduced. The result proves that the proposed algorithm is not only rapidly convergent but also highly accurate. Numerical experiments further show that the new algorithm can significantly improve the computational efficiency and accuracy. Its superiorities will be much more remarkable when it is used to solve larger scale problems. Therefore, it has extensive prospects in the FM-BEM field and other scientific and engineering computing. Full article