Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes for Parallel-in-Time Solvers
<|reference_start|>Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes for Parallel-in-Time Solvers: With the advent of supercomputers, multi-processor environments and parallel-in-time (PinT) algorithms offer ways to solve initial value problems for ordinary and partial differential equations (ODEs and PDEs) over long time intervals, a task often unfeasible with sequential solvers within realistic time frames. A recent approach, GParareal, combines Gaussian Processes with traditional PinT methodology (Parareal) to achieve faster parallel speed-ups. The method is known to outperform Parareal for low-dimensional ODEs and a limited number of computer cores. Here, we present Nearest Neighbors GParareal (nnGParareal), a novel data-enriched PinT integration algorithm. nnGParareal builds upon GParareal by improving its scalability properties for higher-dimensional systems and increased processor count. Through data reduction, the model complexity is reduced from cubic to log-linear in the sample size, yielding a fast and automated procedure to integrate initial value problems over long time intervals. First, we provide both an upper bound for the error and theoretical details on the speed-up benefits. Then, we empirically illustrate the superior performance of nnGParareal, compared to GParareal and Parareal, on nine different systems with unique features (e.g., stiff, chaotic, high-dimensional, or challenging-to-learn systems).<|reference_end|>
@article{gattiglio2024nearest,
    title={Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes
  for Parallel-in-Time Solvers},
    author={Guglielmo Gattiglio, Lyudmila Grigoryeva, Massimiliano Tamborrino},
    journal={arXiv preprint arXiv:2405.12182},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.12182},
    primaryClass={stat.CO cs.DC cs.NA math.NA}
}