Lukas Troska

Michael Griebel, Marc Alexander Schweitzer, Lukas Troska

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.

Michael Griebel, Marc Alexander Schweitzer, Lukas Troska

In this article, we present a parallel discretization and solution method for parabolic problems with a higher number of space dimensions. It consists of a parallel-in-time approach using the multigrid reduction-in-time algorithm MGRIT with its implementation in the library XBraid, the sparse grid combination method for discretizing the resulting elliptic problems in space, and a domain decomposition method for each of the subproblems in the combination method based on the space-filling curve approach. As a result, we obtain an extremely fast and embarrassingly parallel solver with excellent speedup and scale-up qualities, which is perfectly suited for parabolic problems with up to six space dimensions. We describe our new parallel approach and show its superior parallelization properties for the heat equation, the chemical master equation and some exemplary stochastic differential equations.

Zahra Khatami, Lukas Troska, Hartmut Kaiser et al.

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.