This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update & Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy.

翻译：本文介绍了一种用于动态低秩近似的并行、秩自适应矩阵积分器。该方法与先前提出的秩自适应基更新与伽辽金积分器密切相关，但关键区别在于，所有导数方程（包括基方程和伽辽金系数方程）均以并行方式求解。此外，该方法消除了使用增广基矩阵进行潜在高成本系数更新的需求。该积分器还引入了一种新的步长拒绝策略，增强了并行积分器与BUG积分器的鲁棒性。通过构造，该并行积分器继承了BUG积分器和投影分裂积分器的稳健误差界。通过一系列数值实验（涉及辐射传输与放射治疗领域的问题），将并行积分器与BUG积分器进行了对比，结果验证了所提方法的效率。