The inversion of structured sparse matrices is a fundamental yet computationally and memory-intensive task in many scientific applications, such as Bayesian statistical modeling and material science. In certain cases, only particular entries of the full inverse are required. This has motivated the development of so-called selected inversion algorithms (SIA), capable of computing only specific elements of the full inverse. Currently, most SIA implementations are restricted to shared-/distributed-memory CPU architectures or to single GPUs. Here, we introduce novel numerical methods to perform the parallel selected inversion and Cholesky decomposition of positive-definite, block-tridiagonal with arrowhead matrices. A distributed memory, GPU-accelerated implementation of our approach is presented and integrated into the structured solver library Serinv. We demonstrate its performance on synthetic and real datasets from statistical air temperature prediction models and achieve CPU (GPU) speedups of up to$2.6 \times(71.4 \times)$over the SIA of the PARDISO library and up to$14 \times(380.9 \times)$over the MUMPS library, when scaling to 16 processes.

Multivariate Gaussian processes (GPs) offer a powerful probabilistic framework to represent complex interdependent phenomena. They pose, however, significant computational challenges in high-dimensional settings, which frequently arise in spatio-temporal applications. We present DALIA, a highly scalable framework for performing Bayesian inference tasks on spatio-temporal multivariate GPs, based on the methodology of integrated nested Laplace approximations. Our approach relies on a sparse inverse covariance matrix formulation of the GP, puts forward a GPU-accelerated block-dense approach, and introduces a hierarchical, triple-layer, distributed-memory parallel scheme. We showcase weak-scaling performance surpassing the state of the art by two orders of magnitude on a model whose parameter space is 8 × larger and measure strong-scaling speedups of three orders of magnitude when running on 496 GH200 superchips on the Alps supercomputer. Applying DALIA to an air pollution study over northern Italy spanning 48 days, we showcase refined spatial resolutions over the aggregated pollutant measurements.

We present SQUIC , a fast and scalable package for sparse precision matrix estimation. The algorithm employs a second-order method to solve the \(\ell_{1}\) -regularized maximum likelihood problem, utilizing highly optimized linear algebra subroutines. In comparative tests using synthetic datasets, we demonstrate that SQUIC not only scales to datasets of up to a million random variables but also consistently delivers runtimes that are significantly faster than other well-established sparse precision matrix estimation packages. Furthermore, we showcase the application of the introduced package in classifying microarray gene expressions. We demonstrate that by utilizing a matrix form of the tuning parameter (also known as the regularization parameter), SQUIC can effectively incorporate prior information into the estimation procedure, resulting in improved application results with minimal computational overhead.