Parallel Selected Inversion of Block-Tridiagonal with Arrowhead Matrices - Maillou, Vincent; Gaedke-Merzhauser, Lisa; Ziogas, Alexandros Nikolaos; Schenk, Olaf; Luisier, Mathieu (Proceedings / IEEE International Conference on Cluster Computing, 2025)
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 ×(71.4 ×) over the SIA of the PARDISO library and up to 14 ×(380.9 ×) over the MUMPS library, when scaling to 16 processes.