For the Fredholm integral equation u=T u+f on the real line, fast solvers are designed on the basis of a discretized wavelet Galerkin method with the Sloan improvement of the Galerkin solution. The Galerkin system is solved by GMRES or by the Gauss elimination method. Our concept of the fast solver includes the requirements that the parameters of the approximate solution un can be determined in...

The linear flow analysis of turbomachinery aeroelasticity views the unsteady flow as the sum of a background nonlinear flow field and a linear harmonic perturbation. The background state is usually determined by solving the nonlinear steady flow equations. The flow solution representing the amplitude and phase of the unsteady perturbation is instead given by the solution of a large complex line...

The GMRES method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix. GMRES-type methods also have been applied to the solution of linear discrete ill-posed problems. Computational experience indicates that for the latter problems variants of the standard GMRES method, that require the solution to live in the range ...

Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. The algorithm GMRES(m) restarts every m iterations. It is generally thought the information of previous GMRES cycles is lost at the time of a restart, so that each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the rela...

The Generalized Minimal Residual method (GMRES) seeks optimal approximate solutions of linear system Ax = b from Krylov subspaces by minimizing the residual norm ‖Ax − b‖2 over all x in the subspaces. Its main cost is computing and storing basis vectors of the subspaces. For difficult systems, Krylov subspaces of very high dimensions are necessary for obtaining approximate solutions with desire...

GMRES is an iterative method that provides better solutions when dealing with larg linear systems of equations with unsymmetric coefficient matrix. By shifting the Arnoldi process to begin with Ar0 instead of r0, simpler GMRES implementation, proposed by Walker and Zhou in 1994, is obtained that in this method, an upper triangular problem is solved instead of hessenberg least square problem. Th...

When solving large nonsymmetric systems of linear equations with the restarted GMRES algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process. Surprisingly, cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values. Here, two simple examples are presented where GMRES(1) conver...

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of refinement to enhance methods sparse based approximate factorizations. In doing so, we first develop a new error analysis LU- and GMRES-based under general mo...

We describe an immersed boundary method for problems of fluid-solute-structure interaction. The numerical scheme employs linearly implicit timestepping, allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method, and local mesh refinement, making it feasible to resolve the steep gradients associated with the space charge layers as well as t...