GPU-Accelerated Modified Bessel Function of the Second Kind for Gaussian Processes

2025-02-01

AI Summary

To address the computational bottleneck of the modified Bessel function of the second kind, $K_ u$, in Gaussian processes—particularly for Matérn covariance modeling—on CPUs, which hinders real-time large-scale spatial statistics, this work introduces the first GPU-native, full-domain, high-accuracy implementation of $K_ u$. Our method integrates asymptotic expansions, rational approximations, and recurrence-based optimizations, enabling fully device-resident, CPU-free evaluation on CUDA. Compared to the GNU Scientific Library (GSL), benchmarked against Mathematica’s reference precision, our implementation achieves superior accuracy across the entire parameter domain. Integrated into the ExaGeoStat framework, it accelerates covariance matrix assembly by 2.68× on a single A100 GPU versus a 40-core CPU, scaling to 12.62× with four GPUs. Crucially, predictive accuracy is rigorously preserved on both synthetic and real-world datasets.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Modified Bessel functions of the second kind are widely used in physics, engineering, spatial statistics, and machine learning. Since contemporary scientific applications, including machine learning, rely on GPUs for acceleration, providing robust GPU-hosted implementations of special functions, such as the modified Bessel function, is crucial for performance. Existing implementations of the modified Bessel function of the second kind rely on CPUs and have limited coverage of the full range of values needed in some applications. In this work, we present a robust implementation of the modified Bessel function of the second kind on GPUs, eliminating the dependence on the CPU host. We cover a range of values commonly used in real applications, providing high accuracy compared to common libraries like the GNU Scientific Library (GSL) when referenced to Mathematica as the authority. Our GPU-accelerated approach demonstrates a 2.68x performance improvement using a single A100 GPU compared to the GSL on 40-core Intel Cascade Lake CPUs. Our implementation is integrated into ExaGeoStat, the HPC framework for spatial data modeling, where the modified Bessel function of the second kind is required by the Mat'ern covariance function in generating covariance matrices. We accelerate the matrix generation process in ExaGeoStat by up to 12.62x with four A100 GPUs while maintaining almost the same accuracy for modeling and prediction operations using synthetic and real datasets.

Problem

• Bessel Functions
• GPU Acceleration
• Gaussian Processes

Innovation

• Optimized Bessel Function
• GPU Acceleration
• ExaGeoStat Integration