Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Iterative methods for elliptic finite element equations on general meshes

Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included

A study of domain decomposition methods applied to the discretized Helmholtz equation

In this work a domain decomposition based preconditioner of the additive Schwarz type is developed and tested on the linear systems which arise out of the application of the Green\u27s Function/Wave Expansion Discretization (GFD/WED) method to Helmholtz\u27s equation. In order to develop the additive Schwarz preconditioner, use is made of a class of one-sided Artificial Radiation Boundary Conditions (ARBC) developed during the course of this work. These ARBCs are computationally shown to be quite accurate for use on their own. The ARBC\u27s are used to radiatively couple the various sub-domains which are naturally part of domain decomposition based methods in such a manner as to ensure that the system matrix, when restricted to the subdomains, is non-singular. In addition, the inter-domain ARBC is constructed such that the solution to the global linear system is unaffected by the presence of the artificial boundaries. The efficacy and efficiency of the method is demonstrated on one, two, and three-dimensional test cases

Hydrodynamics of Suspensions of Passive and Active Rigid Particles: A Rigid Multiblob Approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. For unbounded suspensions and suspensions sedimented against a single no-slip boundary, we rely on existing analytical expressions for the Rotne-Prager tensor combined with a fast multipole method or a direct summation on a Graphical Processing Unit to obtain an simple yet efficient and scalable implementation. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently-developed rigid-body immersed boundary method to suspensions of freely-moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid body equations converges in a bounded number of iterations regardless of the system size. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.Comment: Under revision in CAMCOS, Nov 201

Preconditioning For Matrix Computation

Preconditioning is a classical subject of numerical solution of linear systems of equations. The goal is to turn a linear system into another one which is easier to solve. The two central subjects of numerical matrix computations are LIN-SOLVE, that is, the solution of linear systems of equations and EIGEN-SOLVE, that is, the approximation of the eigenvalues and eigenvectors of a matrix. We focus on the former subject of LIN-SOLVE and show an application to EIGEN-SOLVE. We achieve our goal by applying randomized additive and multiplicative preconditioning. We facilitate the numerical solution by decreasing the condition of the coefficient matrix of the linear system, which enables reliable numerical solution of LIN-SOLVE. After the introduction in the Chapter 1 we recall the definitions and auxiliary results in Chapter 2. Then in Chapter 3 we precondition linear systems of equations solved at every iteration of the Inverse Power Method applied to EIGEN-SOLVE. These systems are ill conditioned, that is, have large condition numbers, and we decrease them by applying randomized additive preconditioning. This is our first subject. Our second subject is randomized multiplicative preconditioning for LIN-SOLVE. In this way we support application of GENP, that is, Gaussian elimination with no pivoting, and block Gaussian elimination. We prove that the proposed preconditioning methods are efficient when we apply Gaussian random matrices as preconditioners. We confirm these results with our extensive numerical tests. The tests also show that the same methods work as efficiently on the average when we use random structured, in particular circulant, preconditioners instead, but we show both formally and experimentally that these preconditioners fail in the case of LIN-SOLVE for the unitary matrix of discreet Fourier transform, for which Gaussian preconditioners work efficiently