This paper presents a portable, GPU-accelerated implementation of a QR-based singular value computation algorithm in Julia. The singular value decomposition (SVD) is a fundamental numerical tool in scientific computing and machine learning, providing optimal low-rank matrix approximations. Its importance has increased even more in large-scale machine learning pipelines, including large language models (LLMs), where it enables low-rank adaptation (LoRA). The implemented algorithm is based on the classic two-stage QR reduction, consisting of successive matrix reduction to band form and bidiagonal form. Our implementation leverages Julia’s multiple dispatch and metaprogramming capabilities, integrating with the GPUArrays and KernelAbstractions frameworks to provide a unified type and hardware-agnostic function. It supports diverse GPU architectures and data types, and is, to our knowledge, the first GPU-accelerated singular value implementation to support Apple Metal GPUs and half precision. Performance results on multiple GPU backends and data types demonstrate that portability does not require sacrificing performance: the unified function outperforms most linear algebra libraries (MAGMA, SLATE, rocSOLVER, oneMKL) for matrix sizes larger than 1024 × 1024, and achieves 80%-90% of the performance of cuSOLVER for large matrices.

We exploit the widening margin in tensor-core performance between [FP64/FP32/FP16/INT8,FP64/FP32/FP16/FP8/INT8] on NVIDIA [Ampere,Hopper] GPUs to boost the performance of output accuracy-preserving mixed-precision computation of Genome-Wide Association Studies (GWAS) of 305K patients from the UK BioBank, the largest-ever GWAS cohort studied for genetic epistasis using a multivariate approach. Tile-centric adaptive-precision linear algebraic techniques motivated by reducing data motion gain enhanced significance with low-precision GPU arithmetic. At the core of Kernel Ridge Regression (KRR) techniques for GWAS lie compute-bound cubic-complexity matrix operations that inhibit scaling to aspirational dimensions of the population, genotypes, and phenotypes. We accelerate KRR matrix generation by redesigning the computation for Euclidean distances to engage INT8 tensor cores while exploiting symmetry. We accelerate solution of the regularized KRR systems by deploying a new four-precision Cholesky-based solver, which, at 1.805 mixed-precision ExaOp/s on a nearly full Alps system, outperforms the state-of-the-art CPU-only REGENIE GWAS software by five orders of magnitude.

We present a general framework that couples the PaRSEC runtime system and the HiCMA numerical library to solve challenging 3D data-sparse problems. Though formally dense, many matrix operators possess a rank structured property that can be exploited during the most time-consuming computational phase, i.e., the matrix factorization. In particular, this work highlights how a software bundle powered by a task-based programming model can address the heterogeneous workloads engendered by compressing the dense operator. Using Tile Low-Rank (TLR) approximation, our approach consists in capturing the most significant information in each tile of the matrix using a threshold which satisfies the application's accuracy requirements. Matrix operations are performed on the compressed data layout, reducing memory footprint and algorithmic complexity. Our proposed software solution accommodates a range of traditional data structures of linear algebra, i.e., from dense and data-sparse to sparse, within a single matrix operation. Separation of concerns is at the heart: hardware-agnostic implementation, asynchronous execution with a dynamic runtime system, and high performance numerical kernels, to prepare scientific applications to embrace exascale opportunities. This ambition necessitates extensions to PaRSEC that incorporate information related to data structure and rank distribution into the runtime decision-making. We introduce two runtime optimizations to address the challenges encountered when confronted with a large rank disparity: (1) a trimming procedure performed at runtime to cut away data dependencies from the directed acyclic graph discovered to be no longer required after compression and (2) a rank-aware diamond-shaped data distribution to mitigate the load imbalance overheads, reduce data movement, and conserve memory foot-print. We assess our implementation using 3D unstructured mesh deformation based on Radial Basis Function (RBF) interpolation. We report performance results on two different high-performance supercomputers and compare against existing state-of-the-art implementation. Our implementation shows up to 7-fold on Shaheen II and 9-fold on Fugaku performance superiority in situations where the 3D unstructured mesh deformation application renders a matrix operator with low density. Our software framework solves a formally dense 3D problem with 52M mesh points on 65K cores in about half an hour. This multidisciplinary work emphasizes the need for runtime systems to go beyond their primary responsibility of task scheduling on massively parallel hardware system, by synergistically bridging matrix algebra libraries with scientific applications.

We extend the capability of space-time geostatistical modeling using algebraic approximations, illustrating application-expected accuracy worthy of double precision from majority low-precision computations and low-rank matrix approximations. We exploit the mathematical structure of the dense covariance matrix whose inverse action and determinant are repeatedly required in Gaussian log-likelihood optimization. Geostatistics augments first-principles modeling approaches for the prediction of environmental phenomena given the availability of measurements at a large number of locations; however, traditional Cholesky-based approaches grow cubically in complexity, gating practical extension to continental and global datasets now available. We combine the linear algebraic contributions of mixed-precision and low-rank computations within a tile based Cholesky solver with on-demand casting of precisions and dynamic runtime support from PaRSEC to orchestrate tasks and data movement. Our adaptive approach scales on various systems and leverages the Fujitsu A64FX nodes of Fugaku to achieve up to 12X performance speedup against the highly optimized dense Cholesky implementation.

We introduce AL4SAN, a lightweight library for abstracting the APIs of task-based runtime engines. AL4SAN unifies the expression of tasks and their data dependencies. It supports various dynamic runtime systems relying on compiler technology and user-defined APIs. It enables a single application to employ different runtimes and their respective scheduling components, while providing user-obliviousness to the underlying hardware configurations. AL4SAN exposes common front-end APIs and connects to different back-end runtimes. Experiments on performance and overhead assessments are reported on various shared- and distributed-memory systems, possibly equipped with hardware accelerators. A range of workloads, from compute-bound to memory-bound regimes, are employed as proxies for current scientific applications. The low overhead (less than 10 percent) achieved using a variety of workloads enables AL4SAN to be deployed for fast development of task-based numerical algorithms. More interestingly, AL4SAN enables runtime interoperability by switching runtimes at runtime. Blending runtime systems permits to achieve a twofold speedup on a task-based generalized symmetric eigenvalue solver, relative to state-of-the-art implementations. The ultimate goal of AL4SAN is not to create a new runtime, but to strengthen co-design of existing runtimes/applications, while facilitating user productivity and code portability. The code of AL4SAN is freely available at https://github.com/ecrc/al4san, with extensions in progress.