AMG based on compatible weighted matching for GPUs

We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic Multigrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPU. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (spd) matrices, named "coarsening based on compatible weighted matching". It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred as the Suitor algorithm, and show that it allow us to obtain good quality coarse matrices for our AMG on GPUs. We exploit inherent parallelism of modern GPUs in all the kernels involving sparse matrix computations both for the setup of the preconditioner and for its application in a Krylov solver, outperforming preconditioners available in Nvidia AmgX library. We report results about a large set of linear systems arising from discretization of scalar and vector partial differential equations (PDEs).Comment: 11 pages, submitted to the special issue of Parallel Computing related to the 10th International Workshop on Parallel Matrix Algorithms and Applications (PMAA18

Improving Solve Time of aggregation-based adaptive AMG

This paper proposes improving the solve time of a bootstrap AMG designed previously by the authors. This is achieved by incorporating the information, set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both, memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations (PDEs) in both 2d and 3d.Comment: 20 pages, 8 figures, accepted for journal publicatio

Adaptive aggregation on graphs

We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error estimators. We use these estimates to construct aggregation based coarse spaces for graph Laplacians. The estimator is used to assess the quality of an aggregation adaptively. Furthermore, a reshaping algorithm based is tested on several numerical examples.Comment: 17 page

Least Angle Regression Coarsening in Bootstrap Algebraic Multigrid

The bootstrap algebraic multigrid framework allows for the adaptive construction of algebraic multigrid methods in situations where geometric multigrid methods are not known or not available at all. While there has been some work on adaptive coarsening in this framework in terms of algebraic distances, coarsening is the part of the adaptive bootstrap setup that is least developed. In this paper we try to close this gap by introducing an adaptive coarsening scheme that views interpolation as a local regression problem. In fact the bootstrap algebraic multigrid setup can be understood as a machine learning ansatz that learns the nature of smooth error by local regression. In order to turn this idea into a practical method we modify least squares interpolation to both avoid overfitting of the data and to recover a sparse response that can be used to extract information about the coupling strength amongst variables like in classical algebraic multigrid. In order to improve the so-found coarse grid we propose a post-processing to ensure stability of the resulting least squares interpolation operator. We conclude with numerical experiments that show the viability of the chosen approach

An Adaptive Multigrid Method Based on Path Cover

We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic partial differential equations. The PC-$\alpha$AMG is based on unsmoothed aggregation AMG (UA-AMG). To preserve the structure of smooth error down to the coarse levels, we approximate the level sets of the smooth error by first forming vertex-disjoint path cover with paths following the level sets. The aggregations are then formed by matching along the paths in the path cover. In such manner, we are able to build a multilevel structure at a low computational cost. The proposed PC-$\alpha$AMG provides a mechanism to efficiently re-build the multilevel hierarchy during the iterations and leads to a fast nonlinear multilevel algorithm. Traditionally, UA-AMG requires more sophisticated cycling techniques, such as AMLI-cycle or K-cycle, but as our numerical results show, the PC-$\alpha$AMG proposed here leads to nearly optimal standard V-cycle algorithm for solving linear systems with weighted graph Laplacians. Numerical experiments for some real world graph problems also demonstrate PC-$\alpha$AMG's effectiveness and robustness, especially for ill-conditioned graphs

A Posteriori Error Estimates for Solving Graph Laplacians

In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians. In earlier works such estimates were computed by solving global optimization problems, which could be computationally expensive. We propose a novel strategy to compute these estimates by constructing a Helmholtz decomposition on the graph based on a spanning tree and the corresponding cycle space. To compute the error estimator, we solve efficiently the linear system on the spanning tree, and then we solve approximately a least-squares problem on the cycle space. As we show, such an estimator has a nearly-linear computational complexity for sparse graphs under certain assumptions. Numerical experiments are presented to demonstrate the efficacy of the proposed method

Modifying AMG coarse spaces with weak approximation property to exhibit approximation in energy norm

Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation (WAP) property which is necessary and sufficient condition for uniform two-grid convergence. This paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both, PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results

Unified Gas-kinetic Scheme with Multigrid Convergence for Rarefied Flow Study

The unified gas kinetic scheme (UGKS) is a direct modeling method based on the gas dynamical model on the mesh size and time step scales. With the implementation of particle transport and collision in a time-dependent flux function, the UGKS can recover multiple flow physics from the kinetic particle transport to the hydrodynamic wave propagation. In comparison with direct simulation Monte Carlo (DSMC), the equations-based UGKS can use the implicit techniques in the updates of macroscopic conservative variables and microscopic distribution function. The implicit UGKS significantly increases the convergence speed for steady flow computations, especially in the highly rarefied and near continuum regime. In order to further improve the computational efficiency, for the first time a geometric multigrid technique is introduced into the implicit UGKS, where the prediction step for the equilibrium state and the evolution step for the distribution function are both treated with multigrid acceleration. The multigrid implicit UGKS (MIUGKS) is used in the non-equilibrium flow study, which includes microflow, such as lid-driven cavity flow and the flow passing through a finite-length flat plate, and high speed one, such as supersonic flow over a square cylinder. The MIUGKS shows 5 to 9 times efficiency increase over the previous implicit scheme. For the low speed microflow, the efficiency of MIUGKS is several orders of magnitude higher than the DSMC. Even for the hypersonic flow at Mach number 5 and Knudsen number 0.1, the MIUGKS is still more than 100 times faster than the DSMC method for a convergent steady state solution

Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right