Different preconditioning techniques for the iterative method MinRes as solver for the Discrete Sources Method (DSM) are presented. This semi-analytical method is used for light scattering computations by particles in the Mie scattering regime. Its numerical schema includes a linear least-squares problem commonly solved using the QR decomposition method. This could be the subject of numerical d...

We discuss issues related to domain decomposition and multilevel preconditioning techniques which are often employed for solving large sparse linear systems in parallel computations. We introduce a class of parallel preconditioning techniques for general sparse linear systems based on a two level block ILU factorization strategy. We give some new data structures and strategies to construct loca...

This paper presents a preconditioning method based on a recursive multilevel lowrank approximation approach. The basic idea is to recursively divide the problem into two and apply a low-rank approximation to a matrix obtained from the Sherman-Morrison formula. The low-rank approximation may be computed by the partial Singular Value Decomposition (SVD) or it can be approximated by the Lanczos bi...

Krylov subspace methods have become very popular, not only for solving large scale linear systems, but also in the area of model order reduction. It is also well known that the performance of iterative solution techniques depends crucially on the choice of preconditioning matrix. Within the area of model order reduction, the concept of preconditioning has not yet been introduced. However, recen...

Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial diierential equations. The large size and the condition of many technical or physical applications in this area result in the need for eecient par-allelization and preconditioning techniques of the CG method. In particular for very ill-condition...

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi–Davidson approach, where the preconditioned system is restricted to appropriate subspaces of ...

New convergence intervals of parameters αi are derived and applied for solving the modified linear systems, which enables a better understanding of how parameters should be chosen. The convergence theorem for H-matrix is given. Meanwhile, we discuss the convergence results forM -matrices linear systems and give some new preconditioners. Numerical examples are used to illustrate our results. Key...

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution...