Fractional decomposition of matrices and parallel computing

Frédéric Hecht 1 Sidi-Mahmoud Kaber 2
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL - Laboratoire Jacques-Louis Lions
Abstract : We are interested in the design of parallel numerical schemes for linear systems. We give an effective solution to this problem in the following case: the matrix A of the linear system is the product of p nonsingular matrices A m i with specific shape: Ai = I −hiX for a fixed matrix X and real numbers hi. Although having the special form, these matrices Ai arise frequently in the discretization of evolutionary Partial Differential Equations. The idea is to express A −1 as a linear combination of elementary matrices A −k i. Hence the solution of the linear system with matrix A is a linear combination of the solutions of linear systems with matrices A k i. These systems are solved simultaneously on different processors.
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Frédéric Hecht, Sidi-Mahmoud Kaber. Fractional decomposition of matrices and parallel computing. 2018. ⟨hal-01878765⟩

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