Gaussian processes (GPs) are commonly used for geospatial analysis, but they suffer from high computational complexity when dealing with massive data. For instance, the log-likelihood function required in estimating the statistical model parameters for geospatial data is a computationally intensive procedure that involves computing the inverse of a covariance matrix with size n X n, where n represents the number of geographical locations. As a result, in the literature, studies have shifted towards approximation methods to handle larger values of n effectively while maintaining high accuracy. These methods encompass a range of techniques, including low-rank and sparse approximations. Vecchia approximation is one of the most promising methods to speed up evaluating the log-likelihood function. This study presents a parallel implementation of the Vecchia approximation, utilizing batched matrix computations on contemporary GPUs. The proposed implementation relies on batched linear algebra routines to efficiently execute individual conditional distributions in the Vecchia algorithm. We rely on the KBLAS linear algebra library to perform batched linear algebra operations, reducing the time to solution compared to the state-of-the-art parallel implementation of the likelihood estimation operation in the ExaGeoStat software by up to 700X, 833X, 1380X on 32GB GV100, 80GB A100, and 80GB H100 GPUs, respectively. We also successfully manage larger problem sizes on a single NVIDIA GPU, accommodating up to 1M locations with 80GB A100 and H100 GPUs while maintaining the necessary application accuracy. We further assess the accuracy performance of the implemented algorithm, identifying the optimal settings for the Vecchia approximation algorithm to preserve accuracy on two real geospatial datasets: soil moisture data in the Mississippi Basin area and wind speed data in the Middle East.

翻译：高斯过程（GPs）常用于地理空间分析，但在处理大规模数据时面临高计算复杂度的挑战。例如，估计地理空间数据统计模型参数时所需的对数似然函数是一个计算密集型过程，涉及对大小为n×n的协方差矩阵求逆，其中n代表地理位置数量。因此，现有文献已转向近似方法，以在保持高精度的同时有效处理更大的n值。这些方法涵盖低秩近似和稀疏近似等多种技术。Vecchia近似是加速对数似然函数评估的最具前景的方法之一。本研究提出了一种利用当代GPU上的批量矩阵计算实现的Vecchia近似并行方案。该实现依赖批量线性代数例程高效执行Vecchia算法中的各个条件分布。我们采用KBLAS线性代数库执行批量线性代数运算，将ExaGeoStat软件中似然估计运算的当前最优并行实现的求解时间分别降低了700倍（32GB GV100 GPU）、833倍（80GB A100 GPU）和1380倍（80GB H100 GPU）。此外，我们成功在单个NVIDIA GPU上处理了更大规模问题：使用80GB A100和H100 GPU可容纳多达100万个位置点，同时保持必要的应用精度。我们进一步评估了所实现算法的精度性能，为Vecchia近似算法确定了最优设置，以保持两个真实地理空间数据集（密西西比河流域土壤湿度数据和中东地区风速数据）的精度。