Non-invasive multigrid for semi-structured grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid

A Parallel Solver for Graph Laplacians

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm

A matrix-free ILU realization based on surrogates

Matrix-free techniques play an increasingly important role in large-scale simulations. Schur complement techniques and massively parallel multigrid solvers for second-order elliptic partial differential equations can significantly benefit from reduced memory traffic and consumption. The matrix-free approach often restricts solver components to purely local operations, for instance, the Jacobi- or Gauss--Seidel-Smoothers in multigrid methods. An incomplete LU (ILU) decomposition cannot be calculated from local information and is therefore not amenable to an on-the-fly computation which is typically needed for matrix-free calculations. It generally requires the storage and factorization of a sparse matrix which contradicts the low memory requirements in large scale scenarios. In this work, we propose a matrix-free ILU realization. More precisely, we introduce a memory-efficient, matrix-free ILU(0)-Smoother component for low-order conforming finite elements on tetrahedral hybrid grids. Hybrid grids consist of an unstructured macro-mesh which is subdivided into a structured micro-mesh. The ILU(0) is used for degrees-of-freedom assigned to the interior of macro-tetrahedra. This ILU(0)-Smoother can be used for the efficient matrix-free evaluation of the Steklov-Poincare operator from domain-decomposition methods. After introducing and formally defining our smoother, we investigate its performance on refined macro-tetrahedra. Secondly, the ILU(0)-Smoother on the macro-tetrahedrons is implemented via surrogate matrix polynomials in conjunction with a fast on-the-fly evaluation scheme resulting in an efficient matrix-free algorithm. The polynomial coefficients are obtained by solving a least-squares problem on a small part of the factorized ILU(0) matrices to stay memory efficient. The convergence rates of this smoother with respect to the polynomial order are thoroughly studied

Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure

Afivo: a framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods

Afivo is a framework for simulations with adaptive mesh refinement (AMR) on quadtree (2D) and octree (3D) grids. The framework comes with a geometric multigrid solver, shared-memory (OpenMP) parallelism and it supports output in Silo and VTK file formats. Afivo can be used to efficiently simulate AMR problems with up to about $10^{8}$ unknowns on desktops, workstations or single compute nodes. For larger problems, existing distributed-memory frameworks are better suited. The framework has no built-in functionality for specific physics applications, so users have to implement their own numerical methods. The included multigrid solver can be used to efficiently solve elliptic partial differential equations such as Poisson's equation. Afivo's design was kept simple, which in combination with the shared-memory parallelism facilitates modification and experimentation with AMR algorithms. The framework was already used to perform 3D simulations of streamer discharges, which required tens of millions of cells