Algebraic multigrid (AMG) methods are used to approximate solutions to (sparse) linear systems of equations using the multilevel strategy of relaxation and coarse-grid correction that are used in geometric multigrid (GMG) methods. While partial differential equations (PDEs) are often the source of these linear systems, the goal in AMG is to generalize the multilevel process to target problems where the correct coarse problem is not apparent – e.g., unstructured meshes, graph problems, or structured problems where uniform refinement is not effective. In GMG, a multilevel hierarchy is determined from structured coarsening of the problem, followed by defining relaxation and interpolation operators. In contrast, in an AMG method the relaxation method is selected – e.g., Gauss-Seidel – and coarse problems and interpolation are automatically constructed.

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