Geostatistical modeling, one of the prime motivating applications for exascale computing, is a technique for predicting desired quantities from geographically distributed data, based on statistical models and optimization of parameters. Spatial data is assumed to possess properties of stationarity or non-stationarity via a kernel fitted to a covariance matrix. A primary workhorse of stationary spatial statistics is Gaussian maximum log-likelihood estimation (MLE), whose central data structure is a dense, symmetric positive definite covariance matrix of dimension of the number of correlated observations. Two essential operations in MLE are the application of the inverse and evaluation of the determinant of the covariance matrix. These can be rendered through the Cholesky decomposition and triangular solution. In this contribution, we reduce the precision of weakly correlated locations to single-or half-precision based on distance. We thus exploit mathematical structure to migrate MLE to a three-precision approximation that takes advantage of contemporary architectures offering BLAS3-like operations in a single instruction that are extremely fast for reduced precision. We illustrate application-expected accuracy worthy of doubleprecision from a majority half-precision computation, in a context where uniform single precision is by itself insufficient. In tackling the complexity and imbalance caused by the mixing of three precisions, we deploy the PaRSEC runtime system. PaRSEC delivers on-demand casting of precisions while orchestrating tasks and data movement in a multi-GPU distributed-memory environment within a tile-based Cholesky factorization. Application-expected accuracy is maintained while achieving up to 1.59X by mixing FP64/FP32 operations on 1536 nodes of HAWK or 4096 nodes of Shaheen II, and up to 2.64X by mixing FP64/FP32/FP16 operations on 128 nodes of Summit, relative to FP64-only operations, This translates into up to 4.5, 4.7, and 9.1 (mixed) PFlop/s sustained performance, respectively, demonstrating a synergistic combination of exascale architecture, dynamic runtime software, and algorithmic adaptation applied to challenging environmental problems.
This paper highlights the necessary development of new instrumentation tools within the PaRSEC task-based runtime system to leverage the performance of low-rank matrix computations. In particular, the tile low-rank (TLR) Cholesky factorization represents one of the most critical matrix operations toward solving challenging large-scale scientific applications. The challenge resides in the heterogeneous arithmetic intensity of the various computational kernels, which stresses PaRSEC's dynamic engine when orchestrating the task executions at runtime. Such irregular workload imposes the deployment of new scheduling heuristics to privilege the critical path, while exposing task parallelism to maximize hardware occupancy. To measure the effectiveness of PaRSEC's engine and its various scheduling strategies for tackling such workloads, it becomes paramount to implement adequate performance analysis and profiling tools tailored to fine-grained and heterogeneous task execution. This permits us not only to provide insights from PaRSEC, but also to identify potential applications' performance bottlenecks. These instrumentation tools may actually foster synergism between applications and PaRSEC developers for productivity as well as high-performance computing purposes. We demonstrate the benefits of these amenable tools, while assessing the performance of TLR Cholesky factorization from data distribution, communication-reducing and synchronizationreducing perspectives. This tool-assisted performance analysis results in three major contributions: a new hybrid data distribution, a new hierarchical TLR Cholesky algorithm, and a new performance model for tuning the tile size. The new TLR Cholesky factorization achieves an 8× performance speedup over existing implementations on massively parallel supercomputers, toward solving large-scale 3D climate and weather prediction applications.
The task-based programming model associated with dynamic runtime systems has gained popularity for challenging problems because of workload imbalance, heterogeneous resources, or extreme concurrency. During the last decade, lowrank matrix approximations, where the main idea consists of exploiting data sparsity typically by compressing off-diagonal tiles up to an application-specific accuracy threshold, have been adopted to address the curse of dimensionality at extreme scale. In this paper, we create a bridge between the runtime and the linear algebra by communicating knowledge of the data sparsity to the runtime. We design and implement this synergistic approach with high user productivity in mind, in the context of the PaRSEC runtime system and the HiCMA numerical library. This requires to extend PaRSEC with new features to integrate rank information into the dataflow so that proper decisions can be taken at runtime. We focus on the tile low-rank (TLR) Cholesky factorization for solving 3D data-sparse covariance matrix problems arising in environmental applications. In particular, we employ the 3D exponential model of Mateŕn matrix kernel, which exhibits challenging nonuniform high ranks in off-diagonal tiles. We first provide a dynamic data structure management driven by a performance model to reduce extra floating-point operations. Next, we optimize the memory footprint of the application by relying on a dynamic memory allocator, and supported by a rank-aware data distribution to cope with the workload imbalance. Finally, we expose further parallelism using kernel recursive formulations to shorten the critical path. Our resulting high-performance implementation outperforms existing data-sparse TLR Cholesky factorization by up to 7-fold on a large-scale distributed-memory system, while minimizing the memory footprint up to a 44-fold factor. This multidisciplinary work highlights the need to empower runtime systems beyond their original duty of task scheduling for servicing next-generation low-rank matrix algebra libraries.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.