论文标题
基于稀疏网格的自适应光谱Koopman方法
The Sparse-Grid-Based Adaptive Spectral Koopman Method
论文作者
论文摘要
将自适应光谱koopman（ask）方法引入了数值求解自主动力系统，该系统为科学和工程领域的不同领域的众多应用奠定了基础。尽管询问的精度很高，但与runge-kutta这样的传统时间集成方案相比，多维系统的计算在计算上更昂贵。在这项工作中，我们结合了稀疏的网格，并要求加速多维系统的计算。这种基于稀疏的基于网格的ASK（SASK）方法使用Smolyak结构来构建多维搭配点以及相关的多项式，这些多项式用于近似系统的Koopman操作员的特征函数。通过这种方式，与使用张量产品规则相比，搭配点的数量减少了。我们证明，SASK可用于基于其半混凝土形式求解部分微分方程。数值实验表明，sask可以平衡准确性与计算成本，因此提出了加速。
The adaptive spectral Koopman (ASK) method was introduced to numerically solve autonomous dynamical systems that lay the foundation of numerous applications across different fields in science and engineering. Although ASK achieves high accuracy, it is computationally more expensive for multi-dimensional systems compared with conventional time integration schemes like Runge-Kutta. In this work, we combine the sparse grid and ASK to accelerate the computation for multi-dimensional systems. This sparse-grid-based ASK (SASK) method uses the Smolyak structure to construct multi-dimensional collocation points as well as associated polynomials that are used to approximate eigenfunctions of the Koopman operator of the system. In this way, the number of collocation points is reduced compared with using the tensor product rule. We demonstrate that SASK can be used to solve partial differential equations based-on their semi-discrete forms. Numerical experiments illustrate that SASK balances the accuracy with the computational cost, and hence accelerates ASK.
