Fast algebraic multigrid for block-structured dense and Toeplitz-like-plus-Cross systems arising from nonlocal diffusion problems
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Algebraic multigrid (AMG) is one of the most efficient iterative methods for solving large sparse system of equations. However, how to build/check restriction and prolongation operators in practical of AMG methods for nonsymmetric sparse systems is still an interesting open question [Brezina, Manteuffel, McCormick, Runge, and Sanders, SIAM J. Sci. Comput. (2010); Manteuffel and Southworth, SIAM J. Sci. Comput. (2019)]. This paper deals with the block-structured dense and Toeplitz-like-plus-Cross systems, including nonsymmetric indefinite, symmetric positive definite (SPD), arising from nonlocal diffusion problem and peridynamic problem. The simple (traditional) restriction operator and prolongation operator are employed in order to handle such block-structured dense and Toeplitz-like-plus-Cross systems, which is convenient and efficient when employing a fast AMG. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such SPD situations. The numerical experiments are performed in order to verify the convergence with a computational cost of only 𝒪(N N) arithmetic operations, by using few fast Fourier transforms, where N is the number of the grid points. To the best of our knowledge, this is the first contribution regarding Toeplitz-like-plus-Cross linear systems solved by means of a fast AMG.
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