Equation-free modeling aims at extracting low-dimensional macroscopic dynamics from complex high-dimensional systems that govern the evolution of microscopic states. This algorithm relies on lifting and restriction operators that map macroscopic states to microscopic states and vice versa. Combined with simulations of the microscopic state, this algorithm can be used to apply Newton solvers to the implicitly defined low-dimensional macroscopic system or solve it more efficiently using direct numerical simulations. The key challenge is the construction of the lifting and restrictions operators that usually require a priori insight into the underlying application. In this paper, we design an application-independent algorithm that uses diffusion maps to construct these operators from simulation data. Code is available at https://doi.org/10.5281/zenodo.5793299.
Bibliographical noteFunding Information:
Tracy Chin, Jacob Ruth, and Rebecca Santorella were supported by the NSF Grant DMS-1439786 through the Summer@ICERM program. Rebecca Santorella was also supported by the NSF through Grant 1644760. Bjorn Sandstede was supported by the NSF under Grants DMS-1408742, DMS-1714429 and CCF-1740741.
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Bifurcation analysis
- Diffusion maps
- Equation-free modeling