Robust High-Order Projector-Splitting Integrators
Abstract
We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-r approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-r factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.
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BibTeX
@misc{zhang2026robust,
title = {{Robust High-Order Projector-Splitting Integrators}},
author = {Shiheng Zhang and Jingwei Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17157},
eprint = {2608.17157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17157},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Robust High-Order Projector-Splitting Integrators
AU - Shiheng Zhang
AU - Jingwei Hu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.17157
UR - https://arxiv.org/abs/2608.17157
AB - We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-r approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-r factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.
ER -