Gauss-Seidel Smoother Research Articles

A parallel multigrid solver for incompressible flows on computing architectures with accelerators

An efficient parallel multigrid pressure correction algorithm is proposed for the solution of the incompressible Navier–Stokes equations on computing architectures with acceleration devices. The pressure correction procedure is based on the numerical solution of a Poisson-type problem, which is discretized using a fourth-order finite difference compact scheme. Since this is the most time-consuming part of the solver, we propose a parallel pressure correction algorithm using an iterative method based on a block cyclic reduction solution method combined with a multigrid technique. The grid points are numbered with respect to the red–black ordering scheme for the parallel Gauss–Seidel smoother. These parallelization techniques allow the execution of the entire simulation computations on the acceleration device, minimizing memory communication costs. The realization is developed using the OpenACC API, and the numerical method is demonstrated for the solution of two classical incompressible flow test problems. The first is the two-dimensional lid-driven cavity problem over equal mesh sizes while the other is the Stokes boundary layer, which is a decent benchmark problem for unequal mesh spacing. The effect of several multigrid components on modern and legacy acceleration architectures is examined. Eventually the performance investigation demonstrates that the proposed parallel multigrid solver achieves an acceleration of more than 10\(\times \) over the sequential solver and more than 4\(\times \) over multi-core CPU only realizations for all tested accelerators.

A multigrid solver for Stokes control problems

ABSTRACTMultigrid solvers for distributed optimal control problems constrained by Stokes equations are presented. The distributed velocity tracking problem is considered with Dirichlet boundary conditions. The optimality system of the control problem that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods on staggered grids. A coarsening by a factor of three is introduced that results in a nested hierarchy of staggered grids and simplified the intergrid transfer operators. On these grids a distributive Gauss–Seidel smoothing scheme is employed. Numerical experiments are performed to validate the effectiveness and efficiency of the proposed multigrid staggered grid framework.

Efficient relaxed-Jacobi smoothers for multigrid on parallel computers

In this Technical Note, we present a family of Jacobi-based multigrid smoothers suitable for the solution of discretized elliptic equations. These smoothers are based on the idea of scheduled-relaxation Jacobi proposed recently by Yang & Mittal (2014) [18] and employ two or three successive relaxed Jacobi iterations with relaxation factors derived so as to maximize the smoothing property of these iterations. The performance of these new smoothers measured in terms of convergence acceleration and computational workload, is assessed for multi-domain implementations typical of parallelized solvers, and compared to the lexicographic point Gauss–Seidel smoother. The tests include the geometric multigrid method on structured grids as well as the algebraic grid method on unstructured grids. The tests demonstrate that unlike Gauss–Seidel, the convergence of these Jacobi-based smoothers is unaffected by domain decomposition, and furthermore, they outperform the lexicographic Gauss–Seidel by factors that increase with domain partition count.

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

The paper develops a Newton multigrid (MG) method for one- and two-dimensional steady-state shallow water equations (SWEs) with topography and dry areas.It solves the nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs by using Newton's method as the outer iteration and a geometric MG method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton MG method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transitions. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton MG method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

Highly parallel computing of a multigrid solver for 3D Navier–Stokes equations

In order to efficiently obtain all frequencies of the solution, a multigrid solver is used to solve the Navier–Stokes equations for incompressible flows. The method uses a cell-by-cell Gauss–Seidel smoother that is not straightforwardly parallelizable. Moreover the coarsest grids are very coarse and cannot be solved in parallel. The proposed method splits the 3D Cartesian computational domain into well-balanced sub-domains with respect to two dimensions. Efficient parallel procedures using MPI libraries permit us to get a high strong and weak scalability of the whole parallel software. Comparison is done between MPI and hybrid MPI/OpenMP parallelism.

The convection of a Bingham fluid in a differentially-heated porous cavity

Purpose – The purpose of this paper is to determine the manner in which a yield stress fluid begins convecting when it saturates a porous medium. A sidewall-heated rectangular cavity is selected as the testbed for this pioneering work. Design/methodology/approach – Steady solutions are obtained using a second order accurate finite difference method, line relaxation based on the Gauss-Seidel smoother, a Full Approximation Scheme multigrid algorithm with V-cycling and a regularization of the Darcy-Bingham model to smooth the piecewise linear relation between the Darcy flux and the applied body forces. Findings – While Newtonian fluids always convect whenever the Darcy-Rayleigh number is nonzero, Bingham fluids are found to convect only when the Darcy-Rayleigh number exceeds a value which is linearly dependent on both the Rees-Bingham number and the overall perimeter of the rectangular cavity. Stagnation is always found in the centre of the cavity and in regions close to the four corners. Care must be taken over the selection of the regularization constant. Research limitations/implications – The Darcy-Rayleigh number is restricted to values which are at or below 200. Originality/value – This is the first investigation of the effect of yield stress on nonlinear convection in porous media.

High-performance conjugate gradient performance improvement on the K computer

The high-performance conjugate gradient (HPCG) is new benchmark software for supercomputers that provides a more realistic performance metric than existing benchmarks, such as the LINPACK benchmark. The HPCG measures the speed of solving symmetric sparse linear system equations using the conjugate gradient method preconditioned by a multigrid symmetric Gauss–Seidel smoother. The combination of a sparse linear system and a preconditioned conjugate gradient method is widely used in many scientific and engineering computer applications. This study introduces a tuning method for the K computer. According to weak-scaling measurements on the K computer, it has good parallel scalability. Therefore, our tuning strategy focuses on single CPU performance rather than parallel performance. Single CPU performance strongly depends on memory throughput and multicore utilization. Therefore, we attempt to improve memory/cache access performance and multithreading efficiency. As a result, a HPCG score obtained with the K computer achieved second place at SC’14.

Local Multilevel Method on Adaptively Refined Meshes for Elliptic Problems with Smooth Complex Coefficients

AbstractIn this paper, a local multilevel algorithm is investigated for solving linear systems arising from adaptive finite element approximations of second order elliptic problems with smooth complex coefficients. It is shown that the abstract theory for local multilevel algorithm can also be applied to elliptic problems whose dominant coefficient is complex valued. Assuming that the coarsest mesh size is sufficiently small, we prove that this algorithm with Gauss-Seidel smoother is convergent and optimal on the adaptively refined meshes generated by the newest vertex bisection algorithm. Numerical experiments are reported to confirm the theoretical analysis.

Optimizations in a high-performance conjugate gradient benchmark for IA-based multi- and many-core processors

This paper presents optimizations in a high-performance conjugate gradient benchmark (HPCG) for multi-core Intel® Xeon® processors and many-core Xeon Phi™ coprocessors. Without careful optimization, the HPCG benchmark under-utilizes the compute resources available in modern processors due to its low arithmetic intensity and challenges in parallelizing the Gauss–Seidel smoother (GS). Our optimized implementation fuses GS with sparse matrix vector multiplication (SpMV) to address the low arithmetic intensity, overcoming the performance otherwise bound by memory bandwidth. This fusion optimization is progressively more effective in newer generation Xeon processors, demonstrating the usefulness of their larger caches for sparse matrix operations: Sandy Bridge, Ivy Bridge, and Haswell processors achieve 93%, 99%, and 103%, respectively, of the ideal performance with a constraint that matrices are streamed from memory. Our implementation also parallelizes GS using fine-grain level-scheduling, a method that has been believed not to scale with many cores. Our GS implementation scales with 60 cores in Xeon Phi coprocessors, for the finest level of the multi-grid pre-conditioner. At the coarser levels, we address the limited parallelism using block multi-color re-ordering, achieving 21 GFLOPS with one Xeon Phi coprocessor. These optimizations distinguish our HPCG implementation from the others that stream most of the data from main memory and rely on multi-color re-ordering for parallelism. Our optimized implementation has been evaluated in clusters with various configurations, and we find that low-diameter high-radix network topologies such as Dragonfly realize high parallelization efficiencies because of fast all-reduce collectives. In addition, we demonstrate that our optimizations not only benefit the HPCG dataset, which is based on a structured 3D grid, but also a wide range of unstructured matrices.

An efficient algebraic multigrid method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions

SummaryThe quality of the mesh used in the finite element discretizations will affect the efficiency of solving the discreted linear systems. The usual algebraic solvers except multigrid method do not consider the effect of the grid geometry and the mesh quality on their convergence rates. In this paper, we consider the hierarchical quadratic discretizations of three‐dimensional linear elasticity problems on some anisotropic hexahedral meshes and present a new two‐level method, which is weakly independent of the size of the resulting problems by using a special local block Gauss–Seidel smoother, that is LBGS_v iteration when used for vertex nodes or LBGS_m iteration for midside nodes. Moreover, we obtain the efficient algebraic multigrid (AMG) methods by applying DAMG (AMG based on distance matrix) or DAMG‐PCG (PCG with DAMG as a preconditioner) to the solution of the coarse level equation. The resulting AMG methods are then applied to a practical example as a long beam. The numerical results verify the efficiency and robustness of the proposed AMG algorithms. Copyright © 2015 John Wiley & Sons, Ltd.

Nonsymmetric Black Box multigrid with coarsening by three

SUMMARYThe classical Petrov–Galerkin approach to Black Box multigrid for nonsymmetric problems due to Dendy is combined with the recent factor‐three‐coarsening Black Box algorithm due to Dendy and Moulton, along with a powerful symmetric line Gauss–Seidel smoother, resulting in an efficient and robust multigrid solver. Focusing on the convection–diffusion operator, the algorithm is tested and shown to achieve fast and reliable convergence with both first‐order and second‐order accurate upstream discretizations of the convection operator. The solver also exhibits robust behavior with respect to discontinuous jumps in the diffusion coefficient and performs well for recirculating flows over a wide range of diffusion coefficients. The efficiency of the solver is supported by results of an analysis for the case of constant coefficients. Copyright © 2012 John Wiley & Sons, Ltd.

Efficient multicore-aware parallelization strategies for iterative stencil computations

Stencil computations consume a major part of runtime in many scientific simulation codes. As prototypes for this class of algorithms we consider the iterative Jacobi and Gauss-Seidel smoothers and aim at highly efficient parallel implementations for cache-based multicore architectures. Temporal cache blocking is a known advanced optimization technique, which can reduce the pressure on the memory bus significantly. We apply and refine this optimization for a recently presented temporal blocking strategy designed to explicitly utilize multicore characteristics. Especially for the case of Gauss-Seidel smoothers we show that simultaneous multi-threading (SMT) can yield substantial performance improvements for our optimized algorithm on some architectures.

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations – so-called “textbook” multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss–Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations.

Adjoint-based h– p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations

In this paper, we investigate and present an adaptive Discontinuous Galerkin algorithm driven by an adjoint-based error estimation technique for the inviscid compressible Euler equations. This approach requires the numerical approximations for the flow (i.e. primal) problem and the adjoint (i.e. dual) problem which corresponds to a particular simulation objective output of interest. The convergence of these two problems is accelerated by an hp-multigrid solver which makes use of an element Gauss–Seidel smoother on each level of the multigrid sequence. The error estimation of the output functional results in a spatial error distribution, which is used to drive an adaptive refinement strategy, which may include local mesh subdivision ( h-refinement), local modification of discretization orders ( p-enrichment) and the combination of both approaches known as hp-refinement. The selection between h- and p-refinement in the hp-adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values. Numerical results for the inviscid compressible flow over an idealized four-element airfoil geometry demonstrate that both pure h-refinement and pure p-enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach, but requiring fewer degrees of freedom. The proposed hp-adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom, and results in significant savings in computational cost. A high-speed flow test case is used to demonstrate the ability of the hp-refinement approach for capturing strong shocks or discontinuities while improving functional accuracy.

An efficient flexible-order model for 3D nonlinear water waves

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211–228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss–Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.

Robust smoothers for high-order discontinuous Galerkin discretizations of advection–diffusion problems

The multigrid method for discontinuous Galerkin (DG) discretizations of advection–diffusion problems is presented. It is based on a block Gauss–Seidel smoother with downwind ordering honoring the advection operator. The cell matrices of the DG scheme are inverted in this smoother in order to obtain robustness for higher order elements. Employing a set of experiments, we show that this technique actually yields an efficient preconditioner and that both ingredients, downwind ordering and blocking of cell matrices are crucial for robustness.

Multigrid solution method for the steady RANS equations

A novel multigrid method for the solution of the steady Reynolds-averaged Navier–Stokes equations is presented, that gives convergence speeds similar to laminar flow multigrid solvers. The method is applied to Menter’s one-equation turbulence model. New aspects of the method are the combination of nonlinear Gauss–Seidel smoothing on the finest grid with linear coarse-grid corrections, and local damping in the initial stages of the computation, to keep the solution stable; the damping needed is estimated with the nonlinear smoother. Efficiency on the finest grid is increased with full multigrid, second-order accuracy is obtained with defect correction. Tests on boundary layers and airfoil flows show the efficiency of the method.

A Parallel Multigrid Solver for High Frequency Electromagnetic Field Analyses with Small-scale PC Cluster

Finite element analyses of electromagnetic field are commonly used for designing of various electronic devices. The scale of the analyses becomes larger and larger, therefore, a fast linear solver is needed to solve linear equations arising from the finite element method. Since a multigrid solver is the fastest linear solver for these problems, parallelization of a multigrid solver is a quite useful approach. From the viewpoint of industrial applications, an effective usage of a small-scale PC cluster is important due to initial cost for introducing parallel computers.In this paper, a distributed parallel multigrid solver for a small-scale PC cluster is developed. In high frequency electromagnetic field analyses, a special block Gauss-Seidel smoother is used for the multigrid solver instead of general smoothers such as Gauss-Seidel smoother or Jacobi smoother in order to improve a convergence rate. The block multicolor ordering technique is applied to parallelize the smoother. A numerical exsample shows that a 3.7-fold speed-up in computational time and a 3.0-fold increase in the scale of the analysis were attained when the number of CPU was increased from one to five.

A surface capturing method for the efficient computation of steady water waves

A surface capturing method is developed for the computation of steady water–air flow with gravity. Fluxes are based on artificial compressibility and the method is solved with a multigrid technique and line Gauss–Seidel smoother. A test on a channel flow with a bottom bump shows the accuracy of the method and the efficiency of the multigrid solver.

Algebraic multigrid methods for elastic structures with highly discontinuous coefficients

In this paper, we propose two types of algebraic multigrid (AMG) methods to find the numerical solution of elastic structures with highly discontinuous coefficients. One is the interface preserving coarsening AMG method. The idea of this method is to capture the discontinuous behavior of the gradient of the displacement functions along the interfaces. It selects coarse grid points so that all the coarse grids are aligned with the interface for regular interface problems on structured grids and for irregular interface problems on unstructured grids in a purely algebraic way. As a result, AMG with simple linear interpolation and point block Gauss–Seidel smoothing is sufficient to obtain the usual rapid multigrid convergence. The process of coarse grid selection given in this paper can be performed in parallel. Another method introduced in this paper is an AMG method by applying a special block Gauss–Seidel smoother with blocks corresponding to the constant coefficient regions and their interfaces. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the resulting AMG methods are more robust and efficient than the commonly used AMG method in both CPU times and numbers of iteration for elastic structures with highly discontinuous coefficients.