Krylov Iterative Methods Research Articles

Relaxation strategies for nested Krylov methods

This paper studies computational aspects of Krylov methods for solving linear systems where the matrix–vector products dominate the cost of the solution process because they have to be computed via an expensive approximation procedure. In recent years, so-called relaxation strategies for tuning the precision of the matrix–vector multiplications in Krylov methods have proved to be effective for a range of problems. In this paper, we will argue that the gain obtained from such strategies is often limited. Another important strategy for reducing the work in the matrix–vector products is preconditioning the Krylov method by another iterative Krylov method. Flexible Krylov methods are Krylov methods designed for this situation. We combine these two approaches for reducing the work in the matrix–vector products. Specifically, we present strategies for choosing the precision of the matrix–vector products in several flexible Krylov methods as well as for choosing the accuracy of the variable preconditioner such that the overall method is as efficient as possible. We will illustrate this computational scheme with a Schur-complement system that arises in the modeling of global ocean circulation.

Krylov Iterative Methods and the Degraded Effectiveness of Diffusion Synthetic Acceleration for Multidimensional SN Calculations in Problems with Material Discontinuities

A loss in the effectiveness of diffusion synthetic acceleration (DSA) schemes has been observed with certain SN discretizations on two-dimensional Cartesian grids in the presence of material discontinuities. We will present more evidence supporting the conjecture that DSA effectiveness will degrade for multidimensional problems with discontinuous total cross sections, regardless of the particular physical configuration or spatial discretization. Fourier analysis and numerical experiments help us identify a set of representative problems for which established DSA schemes are ineffective, focusing on diffusive problems for which DSA is most needed. We consider a lumped, linear discontinuous spatial discretization of the SN transport equation on three-dimensional, unstructured tetrahedral meshes and look at a fully consistent and a “partially consistent” DSA method for this discretization. The effectiveness of both methods is shown to degrade significantly. A Fourier analysis of the fully consistent DSA scheme in the limit of decreasing cell optical thickness supports the view that the DSA itself is failing when material discontinuities are present in a problem. We show that a Krylov iterative method, preconditioned with DSA, is an effective remedy that can be used to efficiently compute solutions for this class of problems. We show that as a preconditioner to the Krylov method, a partially consistent DSA method is more than adequate. In fact, it is preferable to a fully consistent method because the partially consistent method is based on a continuous finite element discretization of the diffusion equation that can be solved relatively easily. The Krylov method can be implemented in terms of the original SN source iteration coding with only slight modification. Results from numerical experiments show that replacing source iteration with a preconditioned Krylov method can efficiently solve problems that are virtually intractable with accelerated source iteration.

Semi-Toeplitz Preconditioning for the Linearized Navier–Stokes Equations

A Krylov iterative method in combination with a semi-Toeplitz preconditioner to solve the linearized Navier–Stokes equations is presented. A scalar model problem is analyzed showing that the method has very favorable qualities. Numerical experiments for the flow problem corroborate the theory.

Solving Large Scale Semidefinite Programs via an Iterative Solver on the Augmented Systems

The search directions in an interior-point method for large scale semidefinite programming (SDP) can be computed by applying a Krylov iterative method to either the Schur complement equation (SCE) or the augmented equation. Both methods suffer from slow convergence as interior-point iterates approach optimality. Numerical experiments have shown that a diagonally preconditioned conjugate residual method on the SCE typically takes a huge number of steps to converge. However, it is difficult to incorporate cheap and effective preconditioners into the SCE. This paper proposes to apply the preconditioned symmetric quasi-minimal residual (PSQMR) method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior-point iterates. Numerical experiments on SDP problems arising from maximum clique and selected SDPLIB (SDP Library) problems show that moderately accurate solutions can be obtained with a modest number of PSQMR steps using the proposed preconditioned reduced augmented equation. An SDP problem with 127600 constraints is solved in about 6.5 hours to an accuracy of 10-6 in relative duality gap.

A 2D high- β Hall MHD implicit nonlinear solver

A nonlinear, fully implicit solver for a 2D high- β (incompressible) Hall magnetohydrodynamics (HMHD) model is proposed. The task in non-trivial because HMHD supports the whistler wave. This wave is dispersive ( ω∼ k 2) and therefore results in diffusion-like numerical stability limits for explicit time integration methods. For HMHD, implicit approaches using time steps above the explicit numerical stability limits result in diagonally submissive Jacobian systems. Such systems are difficult to invert with iterative techniques. In this study, Jacobian-free Newton–Krylov iterative methods are employed for a fully implicit, nonlinear integration, and a semi-implicit (SI) preconditioner strategy, developed on the basis of a Schur complement analysis, is proposed. The SI preconditioner transforms the coupled hyperbolic whistler system into a fourth-order, parabolic, diagonally dominant PDE, amenable to iterative techniques. Efficiency and accuracy results are presented demonstrating that an efficient fully implicit implementation (i.e., faster than explicit methods) is indeed possible without sacrificing numerical accuracy.

Solution ofthe DiscontinuousP1Equations in Two-Dimensional Cartesian Geometry with Two-Level Preconditioning

We present a new bilinear discontinuous (Galerkin) finite element discretization of the P1 (spherical harmonics) equations, a first order system of equations used for describing neutral particle radiation transport or modeling radiative transfer problems. The discrete equations are described for two-dimensional rectangular meshes; we solve the linear system with Krylov iterative methods. We have developed a novel, two-level preconditioner to improve convergence of the Krylov solvers that is based on a linear continuous finite element discretization of the diffusion equation, solved with a conjugate gradient iteration, preceded and followed by one of several different smoothing relaxations. A Fourier analysis shows that our approach is very effective over a wide range of problems. Numerical experiments confirm the results of the Fourier analysis. Computations for a realistic problem show that the preconditioner is effective and the solution method is efficient in practice.

Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman--Morrison Formula

Let Ax=b be a large, sparse, nonsymmetric system of linear equations. A new sparse approximate inverse preconditioning technique for such a class of systems is proposed. We show how the matrix A0-1 - A-1 , where A0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form $U\Omega V^T$ using the Sherman--Morrison formula. When this factorization process is done incompletely, an approximate factorization may be obtained and used as a preconditioner for Krylov iterative methods. For A0 =sIn, where In is the identity matrix and s is a positive scalar, the existence of the preconditioner for M-matrices is proved. In addition, some numerical experiments obtained for a representative set of matrices are presented. Results show that our approach is comparable with other existing approximate inverse techniques.

Parallel finite element algorithms for the analysis of multiscale plasticity problems

Parallel computing algorithms for the finite element method have been developed to simulate multiscale large deformation plasticity problems. Due to the nature of these problems and the computational methods involved, different parallel computing techniques must be employed for the Taylor-type crystal plasticity (TFE) and the so-called “finite element per single crystal” (FESC) models. For the FESC model, the global stiffness matrix is formed in parallel and a parallel solver employing a number of Krylov iterative methods is used to solve the systems of equations. For the TFE model, a data parallel implementation technique is employed. This technique provides a large reduction in the storage required on a single processor and thus allows for an effective solution of computationally demanding crystal plasticity problems. The efficiency and the performance of the parallel finite element models that have been developed are investigated by comparing simulations performed on an IBM POWER3 SP parallel supercomputer.

Fully Consistent Diffusion Synthetic Acceleration of Linear Discontinuous SN Transport Discretizations on Unstructured Tetrahedral Meshes

We recently presented a method for efficiently solving linear discontinuous discretizations of the two-dimensional P1 equations on rectangular meshes. The linear system was efficiently solved with Krylov iterative methods and a novel two-level preconditioner based on a linear continuous finite element discretization of the diffusion equation. Here, we extend the preconditioned solution method to three-dimensional, unstructured tetrahedral meshes. Solution of the P1 equations forms the basis of a diffusion synthetic acceleration (DSA) scheme for three-dimensional SN transport calculations with isotropic scattering. The P1 equations and the transport equation are both discretized with isoparametric linear discontinuous finite elements so that the DSA method is fully consistent. Fourier analysis in three dimensions and computational results show that this DSA scheme is stable and very effective. The fully consistent method is compared to other “partially consistent” DSA schemes. Results show that the effectiveness of the partially consistent schemes can degrade for skewed or optically thick mesh cells. In fact, one such scheme can degrade to the extent of being unstable even though it is both unconditionally stable and effective on rectangular grids. Results for a model application show that our fully consistent DSA method can outperform the partially consistent DSA schemes under certain circumstances.

Spectral Variants of Krylov Subspace Methods

Krylov iterative methods usually solve an optimization problem, per iteration, to obtain a vector whose components are the step lengths associated with the previous search directions. This vector can be viewed as the solution of a multiparameter optimization problem. In that sense, Krylov methods can be combined with the spectral choice of step length that has recently been developed to accelerate descent methods in optimization. In this work, we discuss different spectral variants of Krylov methods and present encouraging preliminary numerical experiments, with and without preconditioning.

High accuracy solution of three-dimensional biharmonic equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.

Component‐wise algebraic multigrid preconditioning for the iterative solution of stress analysis problems from microfabrication technology

AbstractA methodology for preconditioning discrete stress analysis systems using robust scalar algebraic multigrid (AMG) solvers is evaluated in the context of problems that arise in microfabrication technology. The principle idea is to apply an AMG solver in a segregated way to the series of scalar block matrix problems corresponding to different displacement vector components, thus yielding a block diagonal AMG preconditioner. We study the component‐wise AMG preconditioning in the context of the space decomposition and subspace correction framework [1]. The subspace problems are solved approximately by the scalar AMG solver and the subspace correction is performed either in block diagonal (block Jacobi) or lower triangular (block Gauss–Seidel) fashion. In our test examples we use fully unstructured grids of different sizes. The numerical experiments show robust and efficient convergence of the Krylov iterative methods with component‐wise AMG preconditioning for both 2D and 3D problems. Copyright © 2001 John Wiley & Sons, Ltd.

Highly parallel time integration of viscoelastic flows

A highly parallel time integration method is presented for calculating viscoelastic flows with the DEVSS-G/DG finite element discretization. The method is a synthesis of an operator splitting time integration method that decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. Both steps are performed in parallel. The discontinuous finite element discretization of the hyperbolic constitutive equation leads to highly-parallel element-by-element calculation of the stress at each time step. The Stokes-like problem is solved by using the BiCGStab Krylov iterative method implemented with the block complement and additive levels method (BCALM) preconditioner. The solution method is demonstrated for the calculation of two-dimensional (2D) flow of an Oldroyd-B fluid around an isolated cylinder confined between two parallel plates. These calculations use extremely fine finite elements and expose new features of the solution structure.

A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations

Globalized inexact Newton methods are well suited for solving large-scale systems of nonlinear equations. When combined with a Krylov iterative method, an explicit Jacobian is never needed, and the resulting matrix-free Newton--Krylov method greatly simplifies application of the method to complex problems. Despite asymptotically superlinear rates of convergence, the overall efficiency of a Newton--Krylov solver is determined by the preconditioner. High-quality preconditioners can be constructed from methods that incorporate problem-specific information, and for the incompressible Navier--Stokes equations, classical pressure-correction methods such as SIMPLE and SIMPLER fulfill this requirement. A preconditioner is constructed by using these pressure-correction methods as smoothers in a linear multigrid procedure. The effectiveness of the resulting Newton--Krylov-multigrid method is demonstrated on benchmark incompressible flow problems.

Semi-Implicit Scheme for the DWD Lokal-Modell

The fourth generation of numerical weather prediction (NWP) models is currently under development at the Deutscher Wetterdienst (DWD) consisting of a global grid point model (GME) and limited-area Lokal-Modell (LM). The nonhydrostatic fully compressible LM has been designed to meet high-resol ution regional fore-cast requirements at meso-β and meso-γ scales. The initial LM implementation is based on the NCAR/Penn State MM5 with the addition of a novel generalized terrain-following coordinate and rotated lat-lon grid. A fully 3D semi-implicit time-stepping scheme has been implemented by retaining the full buoyancy term instead of the approximate form found in MM5. In contrast with earlier schemes, mass-lumping is not applied to simplify the elliptic operator on an Arakawa-C/Lorenz grid. The resulting variable-coefficient elliptic problem is solved using a minimal residual Krylov iterative method with line relaxation preconditione rs. The new semi-implicit scheme is compared with a variant of the Klemp–Wilhelmson split-explicit scheme (horizontal explicit, vertical implicit) on the basis of computational efficiency and accuracy at resolutions ranging from 7 km to 400 m. Both idealized 3D mountain wave flows and naturally occuring flows are analyzed. Below the tropopause, the 3D semi-implicit scheme can be more efficient for low Mach number M ≪ 1 flows when the number of small time steps Δt s of the split-explicit approach increases with the sound-speed Courant number.

Time Step Size Selection for Radiation Diffusion Calculations

This work presents efficient solution techniques for radiative transfer in the smoothed particle hydrodynamics discretization. Two choices that impact efficiency are how the material and radiation energy are coupled, which determines the number of iterations needed to converge the emission source, and how the radiation diffusion equation is solved, which must be done in each iteration. The coupled material and radiation energy equations are solved using an inexact Newton iteration scheme based on nonlinear elimination, which reduces the number of Newton iterations needed to converge within each time step. During each Newton iteration, the radiation diffusion equation is solved using Krylov iterative methods with a multigrid preconditioner, which abstracts and optimizes much of the communication when running in parallel. The code is verified for an infinite medium problem, a one-dimensional Marshak wave, and a two and three-dimensional manufactured problem, and exhibits first-order convergence in time and second-order convergence in space. For these problems, the number of iterations needed to converge the inexact Newton scheme and the diffusion equation is independent of the number of spatial points and the number of processors.

A parallel Green's operator for multidimensional quantum scattering calculations

A parallel algorithm for computing multidimensional scattering wave functions is introduced. The inhomogeneous scattering (Lippmann–Schwinger) equation is solved within the discrete variable representation with absorbing boundary conditions, using iterative (Krylov) methods. A parallel Green's operator enables one to distribute the wave function to orthogonal subspaces in which it is processed in parallel. Application to a model problem of electron scattering in a three-dimensional rectangular quantum wire is given. Speedup is demonstrated with an increasing number of processors and with increasing dimensions and/or sampling density. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 167–173, 1998

Parallel Block Preconditioning Techniques for the Numerical Simulation of the Shallow Water Flow Using Finite Element Methods

In this paper, we report our work on applying Krylov iterative methods, accelerated by parallelizable domain-decomposed (DD) preconditioners, to the solution of nonsymmetric linear algebraic equations arising from implicit time discretization of a finite element model of the shallow water equations on a limited-area domain. Two types of previously proposed DD preconditioners are employed and a novel one is advocated to accelerate, with post-preconditioning, the convergence of three popular and competitive Krylov iterative linear solvers. Performance sensitivities of these preconditioners to inexact subdomain solvers are also reported. Autotasking, the parallel processing capability representing the third phase of multitasking libraries on CRAY Y-MP, has been exploited and successfully applied to both loop and subroutine level parallelization. Satisfactory speedup results were obtained. On the other hand, automatic loop-level parallelization, made possible by the autotasking preprocessor, attained only a speedup smaller than a factor of two.