The Method of Moments (MoM) is often e ectively used in the extraction of passive components in modeling integrated circuits and MCM packaging. MoM extraction, however, involves solving a dense system of linear equations, and using direct factorization methods can be prohibitive for large problems. In this paper, we present a Fast Method of Moments Solver (FMMS) for the rapid solution of such l...

A Newton-Krylov unstructured flow solver is developed for higher-order computation of the Euler equations using an upwind scheme. The Generalized Minimal Residual (GMRES) algorithm is used for solving the linear system arising from implicit time discretization of the governing eqautions. An Incomplete Lower-Upper factorization technique is employed as the preconditioning strategy, and an approx...

We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Ax = b. We prove that given the results of a BiCG computation on Ax = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bx = c are identical to those generated by the BiCG computations. Using a unitary equivalence for each of ...

The GMRES method is a popular iterative method for the solution of linear systems of equations with a large nonsymmetric nonsingular matrix. However, little is known about the performance of the GMRES method when the matrix of the linear system is of ill-determined rank, i.e., when the matrix has many singular values of different orders of magnitude close to the origin. Linear systems with such...

The solution of sparse linear systems is the most time-consuming step in running reservoir simulations; over 70% of time is spent on the solution of linear systems derived from the Newton methods [1]. If large highly heterogeneous reservoir models are applied, their linear systems are even harder to solve and require much more simulation time. Hence fast solution techniques are fundamental to l...

We propose a block Krylov subspace version of the GCRO-DR method proposed in [Parks et al. SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented block Krylov subspace. We offer a clean derivation of the method and discuss methods of selecting recycling subspaces at restart as well as implementation decisions in the context of high...

In this paper, we consider both local and global convergence of the Newton algorithm to solve nonlinear problems when GMRES is used to invert the Jacobian at each Newton iteration. Under weak assumptions, we give a suucient condition for an inexact solution of GMRES to be a descent direction in order to apply a backtracking technique. Moreover, we extend this result to a nite diierence scheme c...

This work is the follow-up of the experimental study presented in [3]. It is based on and extends some theoretical results in [15, 18]. In a backward error framework we study the convergence of GMRES when the matrixvector products are performed inaccurately. This inaccuracy is modeled by a perturbation of the original matrix. We prove the convergence of GMRES when the perturbation size is propo...

Recently, the complementary behavior of restarted GMRES has been studied. We observed that successive cycles of restarted block BGMRES (BGMRES(m,s)) can also complement one another harmoniously in reducing the iterative residual. In the present paper, this characterization of BGMRES(m,s) is exploited to form a hybrid block iterative scheme. In particular, a product hybrid block GMRES algorithm ...

We consider the behavior of the GMRES method for solving a linear system Ax = b when A is singular or nearly so, i.e., ill conditioned. The (near) singularity of A may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular A, we give conditions under which the GMRES iterates converge safely to a least-squares solution...