Sequences of linear systems arise in the predictor-corrector method when computing the Pareto front for multi-objective optimization. Rather than discarding information generated when solving one system, it may be advantageous to recycle information for subsequent systems. To accomplish this, we seek to reduce the overall cost of computation when solving linear systems using common recycling methods. In this work, we assessed the performance of recycling minimum residual (RMINRES) method along with a map between coefficient matrices. For these methods to be fully integrated into the software used in Enouen et al. (2022), there must be working version of each in both Python and PyTorch. Herein, we discuss the challenges we encountered and solutions undertaken (and some ongoing) when computing efficient Python implementations of these recycling strategies. The goal of this project was to implement RMINRES in Python and PyTorch and add it to the established Pareto front code to reduce computational cost. Additionally, we wanted to implement the sparse approximate maps code in Python and PyTorch, so that it can be parallelized in future work.

翻译：多目标优化中计算帕累托前沿时，预测-校正法会产生一系列线性系统求解问题。相较于丢弃单个系统求解过程中产生的信息，回收利用这些信息用于后续系统求解可能更具优势。为实现这一目标，我们致力于采用通用回收方法降低线性系统求解的总计算成本。本研究评估了回收最小残差法（RMINRES）及其系数矩阵映射策略的性能。为使这些方法能完全集成到Enouen等人（2022）使用的软件中，需在Python和PyTorch环境中分别完成各方法的可行实现。本文讨论了在计算这些回收策略的高效Python实现过程中遇到的挑战、已采取的解决方案（及部分持续优化中的方案）。本项目旨在用Python和PyTorch实现RMINRES算法，并将其集成至现有帕累托前沿计算框架中以降低计算成本。此外，我们计划在Python和PyTorch中实现稀疏近似映射代码，为后续并行化研究奠定基础。