An algebraic ILU(k) based two-level domain decomposition preconditioner

Abstract

With the ultimate goal of designing a scalable parallel preconditioner for reservoir simulation problems, we combine domain decomposition ideas (prove suitable for parallelization) with incomplete factorizations (which are standard in reservoir simulation) at subdomain level. We introduce an ILU(k)-based two-level domain decomposition preconditioner and compare its performance with a two-level ILU(k)-Block-Jacobi preconditioner.

Implementação paralela de um precondicionador algébrico de dois níveis de decomposição de domínios baseado em ILU(k)

“…O passo 4 constrói o complemento de Schur através de um produto incompleto comoé descrito em [1]. A submatriz A ΓΓé formada pelas interfaces dos subdomínios e portanto…”

“…Conforme descrito em [1], T Jé um operador diagonal, assim podemos construí-lo como um vetor, quéé aplicado através do produto entrada a entrada. O resultadoé então somado globalmente em z Γ .…”

Parallel Implementation of a Two-level Algebraic ILU(k)-based Domain Decomposition Preconditioner

“…This paper discusses the formulation and the parallel implementation of an algebraic ILU(k)based two-level domain decomposition preconditioner first introduced in [2].…”

“…Due to the difficulty in parallelizing ILU(k), it is quite natural to combine ILU(k) and block-Jacobi, so much so that this combination constitutes PETSc's default parallel preconditioner [3]. The algorithm proposed in [2], whose parallel implementation we discuss in this article, seeks to combine the use of (sequential) ILU(K) with two ideas borrowed from domain decomposition methods: (i) the introduction of an interface that connects subdomains, allowing, as opposed to block-Jacobi, for the interaction between subdomains to be taken into account, and (ii) the introduction of a second level, associated to a coarse version of the problem, that speeds up the resolution of low frequency modes. These improvements come at the cost of greater communication, requiring a more involved parallel implementation.…”

“…The main contribution of the two-level preconditioner proposed in [2] is the fine preconditioner, as the coarse level component is a quite simple one. Accordingly, the main contribution in our article is the development of a careful parallel implementation of that fine part; nonetheless, we also take care of the coarse part by proposing low-cost PETSc-based parallel codes for its construction and application.…”