Symmetric linear solvers underpin critical computations in a variety of fields, including climate modeling, engineering, and machine learning, and often rely on computationally intensive processes such as Cholesky decomposition. Vicki Carrica, Rabab Alomairy, and Evelyne Ringoot from the Massachusetts Institute of Technology, along with Alan Edelman, have developed a new portable mixed-precision solver specifically designed to accelerate these calculations on matrix processing units (MXUs). Their work introduces an algorithm that utilizes a hierarchical recursive approach to account for parallelism within the Cholesky decomposition to maximize computational throughput and control numerical precision. By strategically applying lower-precision operations to off-diagonal blocks while maintaining high precision on the diagonal, the team achieved significant speedups on both NVIDIA H200 and AMD MI300X GPUs. This innovation provides up to 27x speedup for symmetric rank k updates and 5x overall speedup for Cholesky decomposition in addition to significant accuracy improvements compared to existing methods.
Research paper summary: “Accelerating linear solves with mixed-precision nested recursive subdivision on AI hardware.”
This paper details a new approach to speed up the solution of linear systems using a combination of mixed-precision arithmetic, nested recursive subdivision, and optimizations for modern AI hardware (GPUs). The core idea is to take advantage of different precision levels and the strengths of recursive algorithms to improve performance and reduce memory usage.
In this study, we introduce an algorithm that utilizes a hierarchical recursive approach to account for parallelism within the Cholesky decomposition to maximize computational throughput and control numerical accuracy. The methodological innovation of this work lies in a custom recursive data structure that strategically allocates low-precision FP16 operations to large off-diagonal blocks while maintaining high precision in diagonal blocks and ensuring numerical stability throughout the computation. This hierarchical recursion increases the granularity of GEMM operations, optimizes hardware utilization on the GPU, and allows fine-grained control over numerical accuracy.
Mixed-precision Cholesky decomposition in MXU
Symmetric linear solvers are important across many scientific and engineering fields, such as climate modeling and structural analysis. This work focuses on Cholesky decomposition utilizing triangular solve (TRSM) and symmetric rank k update (SYRK) as core components.
Scientists achieved a breakthrough by implementing a nested recursive formulation to reveal parallelism through recursive decomposition of TRSM and SYRK subproblems, thereby maximizing computational throughput. Experiments reveal a custom recursive data structure that assigns lower precision FP16 operations to large off-diagonal blocks while maintaining high precision on diagonal blocks to ensure numerical stability. Results show that mixed-precision computation achieves up to 27x speedup on SYRK and 5.3x speedup on TRSM compared to the full-precision baseline. This improved the overall speed of the Cholesky decomposition by a factor of 5.32 compared to cuSOLVER FP64, while simultaneously achieving 100 times more accuracy than the pure FP16 implementation and maintaining 88% of the peak speed. The researchers developed a recursive algorithm for the Cholesky decomposition and its composition operations, a trigonometric solution, and a symmetric rank k update. This maximizes computational throughput and allows fine-grained control over numerical precision.
The main contribution is a custom data structure that allows us to selectively apply lower-precision FP16 operations to off-diagonal blocks while maintaining high precision on diagonal blocks to maintain numerical stability. Specifically, the team achieved up to 27x speedup for symmetric rank K updates and 5x speedup for Cholesky decomposition compared to a full-precision baseline, as well as significant improvements compared to vendor-provided libraries such as cuBLAS and cuSOLVER. Importantly, the mixed-precision approach maintains high accuracy and can achieve 100x better accuracy than a pure FP16 implementation while retaining most of the peak speedup. The authors acknowledge that their detailed performance analysis primarily focuses on NVIDIA and AMD GPU architectures. Future work may extend the solver to other hardware platforms and explore further optimization possibilities within a recursive framework.
