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A new algebraic multigrid solver for semi-structured linear systems

Scientific Achievement

A novel Semi-Structured Algebraic MultiGrid (SSAMG) method has been designed, implemented and optimized utilizing hypre’s SStruct data structures.

Significance and Impact

The new method effectively utilizes the problem’s structure to deliver faster execution times than fully algebraic multigrid (BoomerAMG). Additionally, SSAMG can solve more general problems than hypre’s structured multigrid solver PFMG and has great potential for performing well on GPU architectures.

Research Details

SSAMG uses a multilevel hierarchy built via semi-coarsening and a two-point structured interpolation strategy leading to coarse levels with smaller stencil sizes than unstructured BoomerAMG.
SSAMG has several configuration options allowing for mimicking full-coarsening and switching to BoomerAMG at coarser levels.
The Semi-structured matrix class and linear algebra operations have been redesigned for better performance.

Run times (on Lassen, IBM Power 9) and number of iterations for weak scalability studies of two 3-dimensional Poisson problems comparing BoomerAMG and PFMG (Top only) with SSAMG, all used as preconditioners for conjugate gradient. The arrows in the top grid indicate anisotropies in different directions (diagonal arrows indicate anisotropies in the z-direction).

V. A. Paludetto Magri, R. Falgout, U. M. Yang. A new semi-structured algebraic multigrid. arXiv preprint arXiv:2205.14273.https://arxiv.org/abs/2205.14273

Work was performed at Lawrence Livermore National Laboratory

Problem’s geometry

(2D plane cut)

Weak scalability results

(Top) (Left) Two-dimensional plane cut of a semi-structured grid composed by four parts with different anisotropy directions. Arrows indicate the directions of strong coupling in each part. Diagonal lines stand for anisotropies in the z-direction. (Right) Total execution times (setup + solve) and iteration counts for solving linear systems of increasing sizes in parallel as a function of CPU cores (IBM Power 9). The problem size per core is fixed to 2 million degrees of freedom (weak scaling) and the linear systems are obtained by discretizing the Poisson equation. The new method (SSAMG) provides a 2.8x speedup with respect to hypre’s structured solver PFMG and a 1.6x speedup with respect to an optimized algebraic multigrid configuration (BoomerAMG) in hypre. Since PFMG cannot be applied to semi-structured problems, the problem was modified to one part using the same values as the semi-structured grid.

(Bottom) (Left) Two-dimensional plane cut of a semi-structured grid composed by three parts. (Right) Total execution times (setup + solve) and iteration counts for solving linear systems of increasing sizes in parallel as a function of CPU cores (IBM Power 9). The problem size per core is fixed to 4 million degrees of freedom (weak scaling) and the linear systems are obtained by discretizing the Poisson equation. The new method (SSAMG) provides a 1.6x speedup with respect to an optimized algebraic multigrid configuration (BoomerAMG) in hypre. This problem cannot be modified to a problem that PFMG can be applied to.