The Sparse-Grid-Based Adaptive Spectral Koopman Method

TL;DR

This paper introduces SASK, a method combining sparse grids with the adaptive spectral Koopman approach to efficiently solve multi-dimensional dynamical systems and PDEs, reducing computational costs while maintaining accuracy.

Contribution

The paper presents a novel sparse-grid-based ASK (SASK) method that accelerates computations for multi-dimensional systems by reducing collocation points using the Smolyak structure.

Findings

SASK achieves comparable accuracy to ASK with fewer collocation points.

SASK significantly reduces computational time for multi-dimensional systems.

Numerical experiments confirm the efficiency and accuracy of SASK.

Abstract

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.