GPU-Accelerated Cholesky Factorization of Block Tridiagonal Matrices

Abstract: This paper presents a GPU-accelerated framework for solving block tridiagonal linear systems that arise naturally in numerous real-time applications across engineering and scientific computing. Through a multi-stage permutation strategy based on nested dissection, we reduce the computational complexity from $\mathcal{O}(Nn^3)$ for sequential Cholesky factorization to $\mathcal{O}(\log_2(N)n^3)$ when sufficient parallel resources are available, where $n$ is the block size and $N$ is the number of blocks. The algorithm is implemented using NVIDIA's Warp library and CUDA to exploit parallelism at multiple levels within the factorization algorithm. Our implementation achieves speedups exceeding 100x compared to the sparse solver QDLDL, 25x compared to a highly optimized CPU implementation using BLASFEO, and more than 2x compared to NVIDIA's CUDSS library. The logarithmic scaling with horizon length makes this approach particularly attractive for long-horizon problems in real-time applications. Comprehensive numerical experiments on NVIDIA GPUs demonstrate the practical effectiveness across different problem sizes and precisions. The framework provides a foundation for GPU-accelerated optimization solvers in robotics, autonomous systems, and other domains requiring repeated solution of structured linear systems. The implementation is open-source and available at https://github.com/PREDICT-EPFL/socu.