Leveraging Task-Parallelism with OmpSs in ILUPACK's Preconditioned CG Method

Exploiting task and data parallelism in ILUPACK’s preconditioned CG solver on NUMA architectures and many-core accelerators

“…For example, by reordering and renaming the blocks properly, a block structure similar to (5) is obtained, from which four submatrices can be disassembled: Figure 3 illustrates the dependency tree for the factorization of the diagonal blocks in (9). The edges of the preconditioner directed acyclic graph (DAG) define the dependencies between the diagonal blocks (tasks), i.e., the order in which these blocks of the matrix have to be processed.…”

“…(9), the SpMV and axpylike kernels only operate with the data associated with the leaves of the preconditioner DAGs. For example, for a matrix partitioned as in (9), the SpMV (O1) is decoupled into four matrix-vector products, withĀ 00 ,Ā 11 , A 22 andĀ 33 , which are accumulated as part of the subsequent dot product (O2) [7]. In consequence, the computations with each one of these four leaves in SpMV is fully independent from the others.…”

“…Further concurrency can be exposed by dividing some of these operations into subtasks, as described, for example, at the end of the previous section for the SpMV involving (9).…”

“…In [10] we increased the granularity of the subtasks by modifying the code to merge three pairs of operations in the PCG solve into a single "group" of subtasks each: O1+O2, O3+O4 and O6+O8; see Figure 1. For example, the SpMV+dot in O1+O2, applied to (9), are combined by merging each one of the small matrix-vector productsw i :=Ā ii d k , i = 0, . .…”

“…More recently, in [9] we refurnished our original OpenMP version of ILUPACK, porting it to the OmpSs 4 data-flow task parallel framework. Furthermore, there we introduced task priorities to accelerate the execution of critical tasks and merged small suboperations in the PCG iteration to reduce task management overhead.…”