  This research investigates the implementation mechanism of block-wise ILU(k)
preconditioner on GPU. The block-wise ILU(k) algorithm requires both the level
k and the block size to be designed as variables. A decoupled ILU(k) algorithm
consists of a symbolic phase and a factorization phase. In the symbolic phase,
a ILU(k) nonzero pattern is established from the point-wise structure extracted
from a block-wise matrix. In the factorization phase, the block-wise matrix
with a variable block size is factorized into a block lower triangular matrix
and a block upper triangular matrix. And a further diagonal factorization is
required to perform on the block upper triangular matrix for adapting a
parallel triangular solver on GPU.We also present the numerical experiments to
study the preconditioner actions on different k levels and block sizes.
