[论文解读] Towards a MATLAB Toolbox to compute backstepping kernels using the power series method

Xin Lin, Rafael Vázquez+1 more|arXiv (Cornell University)|Mar 24, 2024

Parallel Computing and Optimization Techniques被引用 5

一句话总结

本文开发了一个基于 MATLAB 的框架，通过幂级数法计算回步核，引入局部幂级数以处理奇点，并展示了相对于符号方法的显著速度提升和收敛性改进。

ABSTRACT

In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This toolbox would exploit MATLAB's linear algebra and sparse matrix manipulation features for enhanced efficiency; our initial findings show considerable improvements in computational speed with respect to the use of symbolical software without loss of precision at high orders. Additionally, we tackle limitations observed in our earlier work, such as slow convergence (due to oscillatory behaviors) and non-converging series (due to loss of analiticity at some singular points). To overcome these challenges, we introduce a technique that mitigates this behaviour by computing the expansion at different points, denoted as localized power series. This approach effectively navigates around singularities, and can also accelerates convergence by using more local approximations. Basic examples are provided to demonstrate these enhancements. Although this research is still ongoing, the significant potential and simplicity of the method already establish the power series approach as a viable and versatile solution for solving backstepping kernel equations, benefiting both novel and experienced practitioners in the field. We anticipate that these developments will be particularly beneficial in training the recently introduced neural operators that approximate backstepping kernels and gains.

研究动机与目标

• 将幂级数方法在回步核计算中的应用推向超越符号工具的水平。
• 开发一个利用稀疏矩阵以提升效率和可扩展性的 MATLAB 框架。
• 通过引入局部幂级数来解决收敛性和解析性问题。
• 提供示例，展示相较于以往工作在速度和精度上的提升。
• 为将其与用于核学习的神经算子方法集成奠定基础。

提出的方法

• 将回步核方程在三角域内以 x 和 xi 的双幂级数来表述。
• 将级数系数转化为 MATLAB 能高效求解的稀疏线性系统。
• 定义算子矩阵，将截断、微分以及边界/迹等操作实现为线性代数运算。
• 通过在所选点处展开核来引入局部幂级数，以绕过奇点并改善收敛性。
• 提供一个框架和实时脚本，以复现基本示例并将性能与符号求解器进行比较。

实验结果

研究问题

• RQ1 是否可以在 MATLAB 中高效实现幂级数方法来计算 PDE 的回步核？
• RQ2 移动展开点（局部幂级数）如何影响奇点附近的收敛性及整体计算时间？
• RQ3 在高阶展开中，MATLAB 稀疏矩阵框架相较于符号工具在性能和精度方面的优势是什么？
• RQ4 如何在实践中扩展该框架以处理多核和耦合的回步问题？

主要发现

• MATLAB 求解器在大截断阶数下相对于 Mathematica 实现了数量级的显著加速，并利用稀疏性提升内存效率。
• 稀疏矩阵表达形式使高阶幂级数核的计算更高效，其随 N 的扩展性比符号方法更好。
• 局部幂级数能有效规避奇点并在原点展开的级数发散时改善收敛性，在所测试的示例中有可证明的收敛性。
• 该方法在保持精度的同时能够使用更大的 N，如在基本示例中与符号解的比较所示。
• 提供 Live Script 以复现结果并将代码适配到新的核方程。
• 该框架有望支持训练近似回步核及增益的神经算子。