Information and Computation

This paper describes and tests a parallel implementation of a factorized approximate inverse preconditioner (FSAI) to accelerate iterative linear system solvers. Such a preconditioner reveals an efficient accelerator of both Conjugate gradient and BiCGstab iterative methods in the parallel solution of large linear systems arising from the discretization of the advection-diffusion equation. The resulting message passing code allows the solution of large problems leading to a very cost-effective algorithm for the solution of large and sparse linear systems.

A new randomized vector algorithm for iterative solution of large linear systems

Multi-scale methods such as wavelet and multigrid schemes are efficient algorithms for the numerical solution of large classes of partial differential equations (PDEs). In this chapter we illustrate the basic ideas behind iterative methods for the solution of large linear systems which arise from the discretisation of PDEs. Besides describing the concepts of wavelet preconditioning, we summarise the main ingredients of multigrid methods. Examples from computational geosciences will show the importance of multi-scale methods for the simulation of large and complex systems.

Accelerated generalized successive overrelaxation method for least squares problems

Many important applications involve the solution of large linear systems with symmetric, but indefinite coefficient matrices. For example, such systems arise in incompressible flow computations and as subproblems in optimization algorithms for linear and nonlinear programs. Existing Krylov-subspace iterations for symmetric indefinite systems, such as SYMMLQ and MINRES, require the use of symmetric positive definite preconditioners, which is a rather unnatural restriction when the matrix itself is highly indefinite with both many positive and many negative eigenvalues. In this note, the authors describe a new Krylov-subspace iteration for solving symmetric indefinite linear systems that can be combined with arbitrary symmetric preconditioners. The algorithm can be interpreted as a special case of the quasi-minimal residual method for general non-Hermitian linear systems, and like the latter, it produces iterates defined by a quasi-minimal residual property. The proposed method has the same work and storage requirements per iteration as SYMMLQ or MINRES, however, it usually converges in considerably fewer iterations. Results of numerical experiments are reported.

Near-circularity for the rational Zolotarev problem in the complex plane

We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

Two variants of modified incomplete block-matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations. Both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations (PDEs), even in the case of quasi-singular problems.

We consider the iterative solution of large linear systems arising from the discretization of nonsymmetric elliptic problems such as convection‐diffusion equations. A preconditioned conjugate gradient (PCG) method is used, based on equivalent preconditioners, whereby a nonsymmetric preconditioning operator with constant coefficients is proposed. Our goal is to study the behaviour of convergence as convection is increasingly dominating. Mesh independent superlinear convergence is derived, and numerical experiments are enclosed.

Efficient vectorizable PDE solvers

Building a scalable and low cost parallel computing environment is becoming a critical matter for scientific computing that requires the repetitive solutions of large linear systems. In this study, a PCI Express based PC cluster is built for the scalable implementation of parallel iterative linear solvers, and the performance and bottlenecks are evaluated. The results show that the communication bandwidth of the internode network is the most important factor for the scalable implementation of parallel sparse linear algebra operations

Finite-Difference Frequency-Domain methods (FDFD) require solution of large linear systems of equations. These large systems are represented by matrix equations including highly sparse coefficient matrices, and they can often only be solved by using iterative methods. This paper presents an algorithm in which the matrix-equation solution approach in an iterative method is replaced by a multi-step solution process. Instead of using a coefficient matrix, the coefficients in the FDFD formulations are kept as three-dimensional arrays, and they are treated as operators. The algorithm is used together with the Bi-Conjugate Gradients Stabilized (BICGSTAB) method. This is applied to a three-dimensional FDFD method to solve for scattering from dielectric objects. It is also applied to two other FDFD methods (a single-grid and a double-grid FDFD) to solve for scattering from chiral objects. It has been shown that the presented algorithm effectively reduces the solution time and memory requirements.

The randomized Kaczmarz (RK) method is an iterative method for approximating the least-squares solution of large linear systems of equations. The standard RK method uses sequential updates, making parallel computation difficult. Here, we study a parallel version of RK where a weighted average of independent updates is used. We analyze the convergence of RK with averaging and demonstrate its performance empirically. We show that as the number of threads increases, the rate of convergence improves and the convergence horizon for inconsistent systems decreases.

A new stride of 4 reduction algorithm is proposed for the solution of large linear systems of tridiagonal equations. The new strategy incorporates a larger granularity in the solution process thus increasing the efficiency of the parallel computation.

Electron-impact excitation and ionization of helium is studied in the $S$-wave model. The problem is treated in full dimensionality using a time-dependent formulation of the exterior complex scaling method that does not involve the solution of large linear systems of equations. We discuss the steps that must be taken to compute stable ionization amplitudes. We present total excitation, total ionization, and single differential cross sections from the ground and $n=2$ excited states and compare our results with those obtained by others using a frozen-core model.