Point-block incomplete LU preconditioning with asynchronous iterations on GPU for multiphysics problems

Point-block matrices arise naturally in multiphysics problems when all variables associated with a mesh point are ordered together, and are different from the general block matrices since the sizes of the blocks are so small one can often invert some of the diagonal blocks explicitly. Motivated by the recent works of Chow and Patel and Chow et al., we propose an efficient incomplete LU (ILU) preconditioner for point-block matrices targeting applications on GPU. The construction of the preconditioner involves two critical steps: (1) the initial guessing of values for the lower and upper triangular matrices; and (2) several sweeps of asynchronous updating of the triangular matrices. Three representative problems are studied to show the advantage of the proposed point-block approach over the standard point-wise approach in terms of the number of GMRES iterations and also the total compute time. Moreover, we compare the proposed algorithm with the level-scheduling based parallel algorithm employed in NVIDIA’s cuSPARSE library as well as the serial method implemented in Intel MKL library, and the experiments show that a 2×–5× speedup can be achieved over the block-based ILU( p) factorizations from the cuSPARSE library.

Convergence versus Divergence Behaviors of Asynchronous Iterations, and Their Applications in Concrete Situations

Asynchronous iterations have long been used in distributed computing algorithms to produce calculation methods that are potentially faster than a serial or parallel approach, but whose convergence is more difficult to demonstrate. Conversely, over the past decade, the study of the complex dynamics of asynchronous iterations has been initiated and deepened, as well as their use in computer security and bioinformatics. The first work of these studies focused on chaotic discrete dynamical systems, and links were established between these dynamics on the one hand, and between random or complex behaviours in the sense of the theory of the same name. Computer security applications have focused on pseudo-random number generation, hash functions, hidden information, and various security aspects of wireless sensor networks. At the bioinformatics level, this study of complex systems has allowed an original approach to understanding the evolution of genomes and protein folding. These various contributions are detailed in this review article, which is an extension of the paper “An update on the topological properties of asynchronous iterations” presented during the Sixth International Conference on Parallel, Distributed, GPU and Cloud Computing (Pareng 2019).