An unconventional robust integrator for dynamical low-rank approximation
An unconventional robust integrator for dynamical low-rank approximation
复制标题
DOI：
10.1007/s10543-021-00873-0
复制
发表时间：
2021-05-10
影响因子：
1.5
通讯作者：
Lubich, Christian
中科院分区：
文献类型：
--
作者：
Ceruti, Gianluca;Lubich, Christian
我们提出并分析了一种数值积分器，该积分器计算大型瞬态矩阵的低秩近似，这些矩阵要么通过增量明确给出，要么是矩阵微分方程的未知解。此外，积分器通过固定多线性秩的 Tucker 张量扩展到时间相关张量的近似。所提出的低秩积分器与用于动态低秩近似的已知投影仪分离积分器不同，但它保留了迄今为止仅针对投影仪分离积分器已知的对小奇异值的重要鲁棒性。与投影仪分离积分器相比，新积分器还提供了一些潜在优势：它避免了投影仪分离积分器的后向时间积分子步骤，这是耗散问题的潜在不稳定子步骤。它提供了更多的并行性，并且当微分方程保留矩阵或张量的对称性或反对称性时，它保留了矩阵或张量的对称性或反对称性。数值实验说明了所提出的积分器的行为。
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.