The paper considers parallel preconditioned iterative methods in Krylov subspaces for solving systems of linear algebraic equations with large sparse symmetric positive-definite matrices resulting from grid approximations of multidimensional problems. For preconditioning, generalized block algorithms of symmetric successive over-relaxation or incomplete factorization type with matching row sums are used. Preconditioners are based on variable-triangular matrix factors with multiple alternations in triangular structure. For three-dimensional grid algebraic systems, methods are based on nested factorizations, as well as on two-level iterative processes. Successive approximations in Krylov subspaces are computed by applying a family of conjugate direction algorithms with various orthogonality and variational properties, including preconditioned conjugate gradient, conjugate residual, and minimal error methods.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 496, 2020, pp. 104–119.
Translated by V. P. Il’in.
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Il’in, V.P. Parallel Variable-Triangular Iterative Methods in Krylov Subspaces. J Math Sci 255, 281–290 (2021). https://doi.org/10.1007/s10958-021-05371-w
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DOI: https://doi.org/10.1007/s10958-021-05371-w