Parallel Acceleration of Krylov Solvers by Factorized Approximate Inverse Preconditioners

Computer Solution of Large Linear Systems

Accelerated generalized successive overrelaxation method for least squares problems

ABSTRACTFour-dimensional variational (4D-Var) data assimilation for operational systems requires the solution of large linear systems that are poorly conditioned in general. In addition to efficient iterative solvers for linear systems, using a good preconditioner is required to guarantee an acceptable solution with a small number of iterations. We consider the assimilation of ocean observations using a weak constraints 4D-Var for the Navy Coastal Ocean Model (NCOM), based on the representer method. Two methods of preconditioning the linear system are implemented, namely the scaling of the linear system by the square root of the observations error variances, and the approximating stabilised representer matrix based on the computation of some representer functions. We evaluate their convergence using criteria such as the norm of the residuals and of the gradient of the cost function, the analysis error evaluated at the observation locations, and finally the convergence of the sequence of analyses. Results from all criteria show that the fastest convergence is achieved with the rescaling preconditioner when the conjugate gradient used to solve the linear system is equipped with a suitable inner product instead of the Euclidean.

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

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.

Projected equation methods for approximate solution of large linear systems

Block SSOR preconditionings for high-order 3D FE systems. II. Incomplete BSSOR preconditionings

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.

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.

Efficient approximate solution of sparse linear systems

Information and Computation

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.

Efficient vectorizable PDE solvers

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.