We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen-Cahn and the Fisher-KPP equation on the manifold of fixed-rank matrices. Furthermore, our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

翻译：我们提出了两种隐式数值格式，用于刚性非线性偏微分方程的低秩时间积分。我们的方法基于Absil、Baker和Gallivan于2007年提出的预条件黎曼信赖域方法。我们在固定秩矩阵流形上求解Allen-Cahn方程和Fisher-KPP方程时，验证了该方法的有效性。此外，我们的方法能够避免通常采用不动点迭代求解内层非线性方程的方法所面临的时间步长限制。最后，我们在Sutti和Vandereycken于2021年提出的相同变分问题上，证明了预条件子的高效性。