Efficient parallel solution to large‐size sparse eigenproblems with block FSAI preconditioning

Abstract

The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for largesize sparse eigenproblems. Although incomplete factorizations with partial fill-in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large-size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi-Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid-dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. (10) where J i b D fl W m.i b 1/ C 1 6 l 6 mi b g and F OEJ i b , W is the rectangular matrix consisting of all rows with index i 2 J i b . It is easy to show that B i b is SPD [15], so an IC decomposition with 800 M. FERRONATO, C. JANNA AND G. PINI BFSAI-IC outperforms FSAI even by a factor larger than 7, whereas with n p > 32, the ratio R progressively approaches 1. Hence, on massively parallel computers with more than 128 processors, the performance of BFSAI-IC and FSAI proves almost equivalent. This should be no surprise as theoretically the limit of BFSAI for n p ! C1 is FSAI itself [15]. Performance with the PJD algorithmAs is known from literature (e.g., [4,7,10,29]), the PJD algorithm can be very efficient and also quite sensitive to the selected combination of user-specified parameters. After several tests, the most Figure 5. Ratio R between the total wall clock time elapsed using FSAI and BFSAI-IC, versus the number of processors n p for computing the s D 10 (left) and s D 40 (right) leftmost eigenpairs with the PJD algorithm. different methods are stopped when the average relative residual of the leftmost s eigenpairs becomes smaller than 10 5 (10 3 for FAULT-639), that is, the highest accuracy that can be achieved

Parallel implementation

“…Methods belonging to the Davidson family try to overcome this limitation by relaxing the precision with which these inverses are approximated (in some cases with a simple preconditioner). As opposed to Krylov methods, general-purpose Davidson solvers are still difficult to find in freely available parallel software, especially with capabilities for non-symmetric and/or generalized problems, despite there being numerous publications developing the methods for different problem types (as §2.1 summarizes) and even describing parallel implementations tailored for certain applications [Heuveline et al, 1997;Nool and van der Ploeg, 2000;Arbenz et al, 2006;Genseberger, 2010;Hwang et al, 2010;Ferronato et al, 2012].…”

A robust adaptive algebraic multigrid linear solver for structural mechanics

“…The SRQCG method is described by Algorithm 3.2. For a practical investigation of this method through numerical experiments based on real-world applications, we refer the reader to [63,64,65].…”

Parallel Jacobi–Davidson with block FSAI preconditioning and controlled inner iterations

“…By distinction, effective parallel eigensolvers can be implemented by using an approximate inverse preconditioning, which is much easier and more efficient to parallelize than ILU. Successful computational experiences with different approximate inverses for the parallel solution of sparse eigenproblems have been reported, for example in , but mainly for symmetric positive definite (SPD) problems.…”

“…The present paper investigates the performance of the JD algorithm with controlled inner iterations and block factorized sparse approximate inverse (Block FSAI) preconditioning in non‐Hermitian eigenproblems. Block FSAI is an efficient parallel preconditioner originally developed for SPD linear systems and eigensolvers , which has been recently generalized to non‐symmetric matrices. The JD method has been implemented for a parallel multi‐CPU computational environment using the message passing interface (MPI) standard.…”