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In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning approach that is applicable to any discretization, i.e. it directly operates on the assembled matrix equations. Moreover, it allows for the effective use of arbitrary processor numbers independent of the dimension of the underlying partial differential equation while maintaining optimal convergence behavior. This is the core property required to attain a sparse grid based combination method with extreme scalability which can utilize exascale parallel systems efficiently. Moreover, this approach provides a basis for the development of a fault-tolerant solver for the numerical treatment of high-dimensional problems. To achieve the required data redundancy we are therefore concerned with large overlaps of our domain decomposition which we construct via space-filling curves. In this paper, we propose our space-filling curve based domain decomposition solver and present its convergence properties and scaling behavior. The results of numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in arbitrary dimension utilizing arbitrary processor numbers., Comment: 28 pages, 11 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2103.03315

We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally sized subproblems and to determine a certain overlap based on the one-dimensional ordering of the space-filling curve. Furthermore we employ agglomeration and a purely algebraic Galerkin discretization in the construction of the coarse problem. This way, the use of $d$-dimensional geometric information is avoided. The subproblems are dealt with in an additive, parallel way, which gives rise to a subspace correction type linear iteration and a preconditioner for the conjugate gradient method. To make the algorithm fault-tolerant we store on each processor, besides the data of the associated subproblem, a copy of the coarse problem and also the data of a fixed amount of neighboring subproblems with respect to the one-dimensional ordering of the subproblems induced by the space-filling curve. This redundancy then allows to restore the necessary data if processors fail during the computation. Theory supports that the convergence rate of such a linear iteration method stays the same in expectation, and only its order constant deteriorates slightly due to the faults. We observe this in numerical experiments for the preconditioned conjugate gradient method in slightly weaker form as well. Altogether, we obtain a fault-tolerant, parallel and efficient domain decomposition method based on space-filling curves which is especially suited for higher-dimensional elliptic problems., Comment: 61 pages, 26 figures

The performance of many parallel applications depends on loop-level parallelism. However, manually parallelizing all loops may result in degrading parallel performance, as some of them cannot scale desirably to a large number of threads. In addition, the overheads of manually tuning loop parameters might prevent an application from reaching its maximum parallel performance. We illustrate how machine learning techniques can be applied to address these challenges. In this research, we develop a framework that is able to automatically capture the static and dynamic information of a loop. Moreover, we advocate a novel method by introducing HPX smart executors for determining the execution policy, chunk size, and prefetching distance of an HPX loop to achieve higher possible performance by feeding static information captured during compilation and runtime-based dynamic information to our learning model. Our evaluated execution results show that using these smart executors can speed up the HPX execution process by around 12%-35% for the Matrix Multiplication, Stream and $2D$ Stencil benchmarks compared to setting their HPX loop's execution policy/parameters manually or using HPX auto-parallelization techniques., Comment: In Proceedings of ESPM2'17: Third International Workshop on Extreme Scale Programming Models and Middleware, Denver, CO, USA, November 12-17,,2017 (ESPM2'17), 8 pages