60 research outputs found
Aggregation-based aggressive coarsening with polynomial smoothing
This paper develops an algebraic multigrid preconditioner for the graph
Laplacian. The proposed approach uses aggressive coarsening based on the
aggregation framework in the setup phase and a polynomial smoother with
sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration
as a preconditioner to the flexible Conjugate Gradient iteration in the solve
phase. We show that by combining these techniques it is possible to design a
simple and scalable algorithm. Results of the algorithm applied to graph
Laplacian systems arising from the standard linear finite element
discretization of the scalar Poisson problem are reported
Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem
This paper introduces bootstrap two-grid and multigrid finite element
approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a
closed surface. The proposed multigrid method is suitable for recovering
eigenvalues having large multiplicity, computing interior eigenvalues, and
approximating the shifted indefinite eigen-problem. Convergence analysis is
carried out for a simplified two-grid algorithm and numerical experiments are
presented to illustrate the basic components and ideas behind the overall
bootstrap multigrid approach
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
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