Function-Based Algebraic Multigrid Method for the 3D Poisson Problem on Structured Meshes
Multilevel methods, such as Geometric and Algebraic Multigrid, Algebraic Multilevel Iteration, Domain Decomposition-type methods have been shown to be the methods of choice for solving linear systems of equations, arising in many areas of Scientific Computing. The methods, in particular the multigrid methods, have been efficiently implemented in serial and parallel and are available via many scientific libraries.
The multigrid methods are primarily used as preconditioners for various Krylov subspace iteration methods. They exhibit convergence that is independent or nearly independent on the number of degrees of freedom and can be tuned to be also robust with respect to other problem parameters. Since these methods utilize hierarchical structures, their parallel implementation might exhibit lesser scalability.
In this work we utilize a different framework to construct multigrid methods, based on an analytical function representation of the matrix, that keeps the amount of computation high and local, and reduces the memory requirements. This approach is particularly suitable for modern computer architectures. An implementation of the latter for the three-dimensional discrete Laplace operator is derived and implemented. The same function representation technology is used to construct smoothers of sparse approximate inverse type.
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