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