Artem Napov

An Algebraic Multigrid Method with Guaranteed Convergence Rate

We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLI-cycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes two- and three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients.

A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization

We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable (HSS) representations. Such matrices appear in many applications, for example, finite-element methods, boundary element methods, and so on. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization, and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUctured Matrices PACKage (STRUMPACK) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.

An Efficient Multicore Implementation of a Novel HSS-Structured Multifrontal Solver Using Randomized Sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to sevenfold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including ...

Algebraic analysis of aggregation-based multigrid

A convergence analysis of two-grid methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M-)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter, which in some sense measures its quality. The procedure is purely algebraic and can be used to control a posteriori the quality of automatic coarsening algorithms. Assuming the aggregation pattern is sufficiently regular, it is further shown that the resulting bound is asymptotically sharp for a large class of elliptic boundary value problems, including problems with variable and discontinuous coefficients. In particular, the analysis of typical examples shows that the convergence rate is insensitive to discontinuities under some reasonable assumptions on the aggregation scheme. Copyright

A massively parallel solver for discrete Poisson-like problems

The paper considers the parallel implementation of an algebraic multigrid method. The sequential version is well suited to solve linear systems arising from the discretization of scalar elliptic PDEs. It is scalable in the sense that the time needed to solve a system is (under known conditions) proportional to the number of unknowns. The associate software code is also robust and often significantly faster than other algebraic multigrid solvers. The present work addresses the challenge of porting it on massively parallel computers. In this view, some critical components are redesigned, in a relatively simple yet not straightforward way. Thanks to this, excellent weak scalability results are obtained on three petascale machines among the most powerful today available. Efficient solution method for discrete Poisson like problems on petascale computers.Excellent weak scalability for a software that is among the best in sequential mode.Innovative multigrid design.

Smoothing factor, order of prolongation and actual multigrid convergence

We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V-cycle multigrid. We derive a two-sided bound that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow and away from 1 when both the smoothing factor and an additional parameter are small enough. Besides the smoothing factor, the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix associated with small eigenvalues. Nice V-cycle convergence is guaranteed if the tangent of this angle has an upper bound proportional to the eigenvalue, whereas nice two-grid convergence requires a bound proportional to the square root of the eigenvalue. We also discuss the well-known rule which relates the order of the prolongation to that of the differential operator associated to the problem. We first define a frequency based order which in most cases amounts to the so-called high frequency order as defined in Hemker (J Comput Appl Math 32:423–429, 1990). We give a firmer basis to the related order rule by showing that, together with the requirement of having the smoothing factor away from one, it provides necessary and sufficient conditions for having the two-grid convergence rate away from 1. A stronger condition is further shown to be sufficient for optimal convergence with the V-cycle. The presented results apply to rigorous Fourier analysis for regular discrete PDEs, and also to local Fourier analysis via the discussion of semi-positive systems as may arise from the discretization of PDEs with periodic boundary conditions.

An algebraic multifrontal preconditioner that exploits the low-rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off-diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off-diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver.

When does two-grid optimality carry over to the V-cycle?

We investigate additional condition(s) that confirm that a V-cycle multigrid method is satisfactory (say, optimal) when it is based on a two-grid cycle with satisfactory (say, level-independent) convergence properties. The main tool is McCormicks bound on the convergence factor (SIAM J. Numer. Anal. 1985; 22:634–643), which we showed in previous work to be the best bound for V-cycle multigrid among those that are characterized by a constant that is the maximum (or minimum) over all levels of an expression involving only two consecutive levels; that is, that can be assessed considering only two levels at a time. We show that, given a satisfactorily converging two-grid method, McCormicks bound allows us to prove satisfactory convergence for the V-cycle if and only if the norm of a given projector is bounded at each level. Moreover, this projector norm is simple to estimate within the framework of Fourier analysis, making it easy to supplement a standard two-grid analysis with an assessment of the V-cycle potentialities. The theory is illustrated with a few examples that also show that the provided bounds may give a satisfactory sharp prediction of the actual multigrid convergence. Copyright

An efficient multigrid method for graph Laplacian systems II: robust aggregation

We consider the iterative solution of linear systems whose matrices are Laplacians of undirected graphs. Designing robust solvers for this class of problems is challenging due to the diversity of connectivity patterns encountered in practical applications. Our starting point is a recently proposed aggregation-based algebraic multigrid method that combines the recursive static elimination of the vertices of degree 1 with the degree-aware rooted aggregation (DRA) algorithm. The latter always produces aggregates big enough to ensure that the preconditioner cost per iteration is low. Here we further improve the robustness of the method by controlling the quality of the aggregates. More precisely, “bad” vertices are removed from the aggregates formed by the DRA algorithm until a quality test is passed. This ensures that the two-grid condition number is nicely bounded, whereas the cost per iteration is kept low by reforming too small aggregates when it happens that the mean aggregate size is not large enough. T...

A divide-and-conquer bound for aggregate’s quality and algebraic connectivity

Abstract We establish a divide-and-conquer bound for the aggregate’s quality and algebraic connectivity measures, as defined for weighted undirected graphs. Aggregate’s quality is defined on a set of vertices and, in the context of aggregation-based multigrid methods, it measures how well this set of vertices is represented by a single vertex. On the other hand, algebraic connectivity is defined on a graph, and measures how well this graph is connected. The considered divide-and-conquer bound for aggregate’s quality relates the aggregate’s quality of a union of two disjoint sets of vertices to the aggregate’s quality of the two sets. Likewise, the bound for algebraic connectivity relates the algebraic connectivity of the graph induced by a union of two disjoint sets of vertices to the algebraic connectivity of the graphs induced by the two sets.