GMRES Solver Research Articles

Optimization of power consumption in the iterative solution of sparse linear systems on graphics processors

In this paper, we analyze the power consumption of different GPU-accelerated iterative solver implementations enhanced with energy-saving techniques. Specifically, while conducting kernel calls on the graphics accelerator, we manually set the host system to a power-efficient idle-wait status so as to leverage dynamic voltage and frequency control. While the usage of iterative refinement combined with mixed precision arithmetic often improves the execution time of an iterative solver on a graphics processor, this may not necessarily be true for the power consumption as well. To analyze the trade-off between computation time and power consumption we compare a plain GMRES solver and its preconditioned variant to the mixed-precision iterative refinement implementations based on the respective solvers. Benchmark experiments conclusively reveal how the usage of idle-wait during GPU-kernel calls effectively leverages the power-tools provided by hardware, and improves the energy performance of the algorithm.

An analysis for the DIIS acceleration method used in quantum chemistry calculations

This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and Hartree–Fock calculations and in the Coupled Cluster method. Taking up results from Harrison (J Comput Chem 25:328, 2003), we show that for the general nonlinear case, DIIS corresponds to a projected quasi-Newton/secant method. For linear systems, we establish connections to the well-known GMRES solver and transfer according (positive as well as negative) convergence results to DIIS. In particular, we discuss the circumstances under which DIIS exhibits superlinear convergence behaviour. For the general nonlinear case, we then use these results to show that a DIIS step can be interpreted as step of a quasi-Newton method in which the Jacobian used in the Newton step is approximated by finite differences and in which the according linear system is solved by a GMRES procedure, and give according convergence estimates.

Reordering and incomplete preconditioning in serial and parallel adaptive mesh refinement and coarsening flow solutions

SUMMARYThe effects of reordering the unknowns on the convergence of incomplete factorization preconditioned Krylov subspace methods are investigated. Of particular interest is the resulting preconditioned iterative solver behavior when adaptive mesh refinement and coarsening (AMR/C) are utilized for serial or distributed parallel simulations. As representative schemes, we consider the familiar reverse Cuthill–McKee and quotient minimum degree algorithms applied with incomplete factorization preconditioners to CG and GMRES solvers. In the parallel distributed case, reordering is applied to local subdomains for block ILU preconditioning, and subdomains are repartitioned dynamically as mesh adaptation proceeds. Numerical studies for representative applications are conducted using the object‐oriented AMR/C software system libMesh linked to the PETSc solver library. Serial tests demonstrate that global unknown reordering and incomplete factorization preconditioning can reduce the number of iterations and improve serial CPU time in AMR/C computations. Parallel experiments indicate that local reordering for subdomain block preconditioning associated with dynamic repartitioning because of AMR/C leads to an overall reduction in processing time. Copyright © 2011 John Wiley & Sons, Ltd.

Partitioned solvers for the fluid-structure interaction problems with a nearly incompressible elasticity model

In this paper, we present some analysis and numerical studies on two partitioned fluid-structure interaction solvers: a preconditioned GMRES solver and a Newton based solver, for the fluid-structure interaction problems employing a nearly incompressible elasticity model in a classical mixed displacement-pressure formulation. Both are highly relying on robust and efficient solvers for the fluid and structure sub-problems obtained from an extended and stabilized finite element discretization on hybrid meshes. A special algebraic multigrid method capable of handling such general saddle point systems for the incompressible and nearly incompressible models is investigated.

Applying wavelets to improve the boundary element method for elasticity problems

This paper describes application of fast wavelet transforms in the boundary element method to solve 2D elasticity problems. Daubechies compactly supported orthogonal wavelets have been applied to compress dense and fully populated matrices arising from BEM. GMRES solver is then used to solve linear algebraic systems. A comprehensive sensitivity study is presented to answer the questions like, which order and level of D-wavelets and thresholding parameters are efficient for elasticity problems. Numerical results include a precise study on effect of applying different wavelet orders (D4, D6, D8, D10 and D12), levels and thresholds on the solution accuracy for displacements and stresses and compression ratio of sparsified matrices. The suitable order, level and thresholding parameter as well as saving in computer time and memory are presented for practical engineering problems. The results show that the proposed method is efficient for large problems.

An Adaptive Fast Multipole Higher Order Boundary Element Method for Power Frequency Electric Field of Substation

An adaptive fast multipole higher order boundary element method combining fast multipole (FM) with a higher order boundary element method is studied to solve the power frequency electric field (PFEF) of substations. In this new technique, the iterative equation solver GMRES is used in the FM, where matrix-vector multiplications are calculated using fast multipole expansions. The coefficients in the preconditioner for GMRES are stored and are used repeatedly in the direct evaluations of the near-field contributions. Then a 500kV outdoor substation is modeled and the PFEF of the substation is analyzed by the novel algorithm and other conventional methods. The results show that, in computational cost and the storages capability aspects, the algorithm proposed in this study has obvious advantages. It is suitable for the calculation of the large-scale PFEF in complex substations and the design of electromagnetic compatibility.

A BEM/FEM Coupling Approach for Fluid-Structure Interaction Simulation of Cell Motion

In this paper, accurate and efficient simulation of cell motion in a biological fluid flow is investigated. The membrane of a moving cell is represented by a thin shell composed of incompressible neo-Hookean elastic materials and the liquids around the membrane are approximated as incompressible Newtonian flows with low Reynolds numbers. The biofluid mechanics is approximated by the Stokes flow equations. A low-order BEM model is developed for the two biological fluids coupled at the membrane surface. The moving boundary problem in fluid mechanics can be effectively solved using the BEM with a GMRES solver. The FEM model based on a flat thin shell element is further developed to predict the membrane load due to the large deformation of a moving cell. Computational efficiency is greatly improved due to the one-dimensional reduction in the present BEM and FEM models. The BEM solver for the biological fluids is coupled with the FEM solver for the cell membrane at the membrane surface. The position of themembrane surface nodes is advanced in time by using the classical fourth-order Runge-Kutta method. Numerical instability is avoided by using a relatively small time step. Further numerical instabilities in the FEM solver is alleviated by using various techniques. The present method is applied to the FSI problems of cell motion in a cylindrical flow. Numerical examples can illustrate the distinct accuracy, efficiency and robustness of the present method. Furthermore, the importance of bending stiffness of a cell membrane for stable cell motion simulation is emphasized. It is suggested that the present approach be an appealing alternative for simulating the fluid-structure interaction of moving cells. AMS subject classifications: 74K25, 74S15, 74S05, 74F10 PACS: 02.70.Dh, 02.30.Rz, 46.70.De, 83.10.-y

Fast Integral Equation Solution Techniques for Planar-3D Structures in Multilayered Media

In this contribution, a hybrid space/spectral domain integral equation approach is presented for the characterization of quasi-3D structures in multilayered media. The verti- cal conductors for the quasi-3D components like vias, through holes, airbridges, flnite dielectric regions etc. can cross an arbitrary number of dielectric layers with arbitrary vertical discretiza- tion. The electrodynamic behavior of the vertical currents is incorporated by extended spectral domain Green's functions derived by analytical integrations over the vertical direction and by a summation over the Cartesian wavenumbers which must be computed only once and stored in a database. For short lateral distances, the couplings are computed with a subtraction tech- nique based on asymptotic representations of the extended Green's functions, whereas for larger distances and for group coupling computations in context with fast matrix vector product evalu- ations, adaptive integration path deformations with extended Laguerre quadrature methods are applied. For the solution of the linear systems of equations, a GMRES solver was implemented together with a diakoptic preconditioner based on the group decomposition of larger structures, combined with a reversed Cuthill McKee reordering applied to the submatrix pattern of the system matrix. With all these techniques, the analysis of complex quasi-3D structures can be carried out with the same performance than pure planar microstrip and/or slotline structures.

A fast BEM for the analysis of damaged structures with bonded piezoelectric sensors

A fast boundary element method for the analysis of three-dimensional solids with cracks and adhesively bonded piezoelectric patches, used as strain sensors, is presented. The piezoelectric sensors, as well as the adhesive layer, are modeled using a 3D state-space finite element approach. The piezoelectric patch model is formulated taking into account the full electro-mechanical coupling and embodying the suitable boundary conditions and it is eventually expressed in terms of the interface variables, to allow a straightforward coupling with the underlying host structure, which is modeled through a 3D dual boundary element method, for accurate analysis of cracks. The technique is computationally enhanced, in terms of memory storage and solution time, using the hierarchical format in conjunction with a GMRES solver. An original strategy retaining the advantages of the fast hierarchical solution without increasing the implementation complexity to take into account the piezoelectric patches is proposed for the solution of the final system. The presented work is a step towards modeling of structural health monitoring systems.

Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid ( ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton–Krylov method. The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton–GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem. It is concluded that the Newton–GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.

Fast Solution of a 3-D Integral Model for the Analysis of ITER Superconducting Coils

In this paper, we simulate the electromagnetic behavior of cable-in-conduit conductors (CICCs) and coils using a 3-D integral formulation of magneto-quasistatic Maxwell's equations, where the superconductors are represented by a power-law. The integral equation is numerically formulated using edge-elements and the nonlinear problem is discretized with a quasi-Newton algorithm. The solution of each iteration step is achieved with an iterative GMRES solver, which benefits from an singular value decomposition of the full matrix representing the integral operator. The model is applied to study the electromagnetic behavior of international thermonuclear experimental reactor (ITER) CICC and coils.

A Preconditioned Recycling GMRES Solver for Stochastic Helmholtz Problems

We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients. Karhunen-Loeve expansions are used to represent the stochastic variables and the stochastic Galerkin method with dou- ble orthogonal polynomials is used to derive a sequence of uncoupled deterministic equations. We show numerically that the Schwarz preconditioned recycling GMRES method is an effective technique for solving the entire family of linear systems and, in particular, the use of recycled Krylov subspaces is the key element of this successful approach.

Parallel solution of large-scale free surface viscoelastic flows via sparse approximate inverse preconditioning

Though computational techniques for two-dimensional viscoelastic free surface flows are well developed, three-dimensional flows continue to present significant computational challenges. Fully coupled free surface flow models lead to nonlinear systems whose steady states can be found via Newton’s method. Each Newton iteration requires the solution of a large, sparse linear system, for which memory and computational demands suggest the application of an iterative method, rather than the sparse direct methods widely used for two dimensional simulations. The Jacobian matrix of this system is often ill-conditioned, resulting in unacceptably slow convergence of the linear solver; hence preconditioning is essential. We propose a variant sparse approximate inverse preconditioner for the Jacobian matrix that allows for the solution of problems involving more than a million degrees of freedom in challenging parameter regimes. Construction of this preconditioner requires the solution of small least squares problems that can be simply parallelized on a distributed memory machine. The performance and scalability of this preconditioner with the GMRES solver are investigated for two- and three-dimensional free surface flows on both structured and unstructured meshes in the presence and absence of viscoelasticity. The results suggest that this preconditioner is an extremely promising candidate for solving large-scale steady viscoelastic flows with free surfaces.

A Rigid-fiber-based Boundary Element Model for Strength Simulation of Carbon Nanotube Reinforced Composites

Summary Carbon nanotubes (CNTs) may provide ultimate enhancement in stiffness and strength for composite materials. This paper presents a rigid-fiber-based boundary integral equation formulation for the numerical simulation of debonding process and the corresponding strength of CNT reinforced composites. The CNT/matrix interfaces are assumed to fail when the interfacial shear force reaches a prescribed threshold, and the CNTs and matrix are considered to be detached in the failed areas. The matrix with one or several tens of originally well-bonded CNTs is subjected to an incremental tensile load and the effective stress-strain relations are readily obtained by the introduction of CNT/matrix debonding processes. An equivalent strength of CNT composites is also defined to study the effect of fracture process. In order to analyze considerable CNTs by use of the fast multipole method, a rigid-fiber-related preconditioning technique is introduced to deal with the case of CNT/matrix detachment. The boundary element model is solved on a desktop computer by using both the traditional GMRES solver and the fast multipole method. The impact of several micro-structural parameters on the debonding process and strength of CNT reinforced composites is discussed in the numerical tests, and some results are compared with experimental ones reported in the literature.

An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions

AbstractThe smoothed particle hydrodynamics (SPH) method has proven useful for modeling large deformation of fluids including fluids with stress‐free surfaces. Because of the Lagrangian nature of the method, it is well suited to address the thermal evolution of these free surface flows. Boundary conditions at the interface of the fluid with a solid wall are usually enforced through the use of boundary particles. However, applying conditions at free surfaces, in particular gradient boundary conditions, can be problematic with traditional SPH formulations due to the degradation of the gradient approximation in these regions. Compounding this difficulty is that traditional approximations of the Laplacian operator suffer a similar degradation near free surfaces. A new SPH formulation of the Laplacian operator is presented, which improves the accuracy near free surface boundaries. This new form is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems, but includes higher‐order terms in the appropriate Taylor series. Comparisons with other approximations of second‐order derivatives are given. The discretization is tested by solving steady‐state and transient problems of thermal diffusion using the Backward Euler method with a GMRES solver. Boundary conditions are imposed through an augmented matrix. Copyright © 2008 John Wiley & Sons, Ltd.

On the evaluation of layer potentials close to their sources

When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation. This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplace’s equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries.

The multigrid preconditioned flexible GMRES solver for hierarchical TVFEM analysis

AbstractThe H1(curl) conforming finite element method is applied for the finite‐element modeling of electromagnetic structures. A p‐version multigrid method combined with flexible GMRES solver is introduced to solve the finite element system. In this scheme, the p‐version multigrid is applied as the preconditioner and the flexible GMRES solver is performed as outer iteration. Therefore, the proposed technique can efficiently eliminate the low frequency components of the error of the numerical solution and can achieve fast convergence. Numerical simulation demonstrates the superior numerical convergence of the proposed method. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2012–2018, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22577

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457–492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier–Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton–Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

Application of single-level, pointwise algebraic, and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.

An implicit high order finite volume scheme for the solution of 3D Navier–Stokes equations with new discretization of diffusive terms

The computation of three-dimensional viscous flows on complex geometries requiring distorted meshes is of great interest. This paper presents a finite volume solver using a quadratic reconstruction of the unknowns for the advective fluxes computation, and a conservative and consistent discretization of the diffusive terms, based on an extended version of the Coirier's diamond path. A fully implicit time integration procedure is employed with a preconditioned matrix-free GMRES solver.