In this work we present a novel Quasi-Newton technique for the black-box partitioned coupling of interface coupled problems. The new RandomiZed Multi-Vector Quasi-Newton method stems from the combination of the original Multi-Vector Quasi-Newton technique with the randomized Singular Value Decomposition algorithm, avoiding thus any dense DOFs-sized square matrix operation. This results in a reduction from quadratic to linear complexity in terms of the number of DOFs. Besides this, the need of storing the old inverse Jacobian is also avoided. Instead, only two very “thin” matrices are required to be saved, thus implying a much smaller memory footprint. Furthermore, our proposal can be used free of any user-defined parameter. The article describes the application of the method to the FSI interface residual equations in both Interface Quasi-Newton and Interface Block Quasi-Newton forms. For the latter, we also derive a closed form expression for the update, thus avoiding any linear system of equations resolution, by applying the Woodbury matrix identity to the inverse Jacobian decomposition matrices.

在这项工作中，我们提出了一种新的拟牛顿技术，用于接口耦合问题的黑盒分区耦合。新的 RandomiZed Multi-Vector Quasi-Newton 方法源于原始的 Multi-Vector Quasi-Newton 技术与随机奇异值分解算法的结合，从而避免了任何密集的 DOFs 大小的方阵操作。这导致在 DOF 数量方面从二次复杂性降低到线性复杂性。除此之外，还避免了存储旧的逆雅可比行列式的需要。相反，只需要保存两个非常“薄”的矩阵，因此意味着内存占用要小得多。此外，我们的提议可以在没有任何用户定义参数的情况下使用。本文介绍了该方法在界面拟牛顿和界面块拟牛顿形式的 FSI 界面残差方程中的应用。对于后者，我们还通过将伍德伯里矩阵恒等式应用于逆雅可比分解矩阵，推导了更新的封闭形式表达式，从而避免了任何线性方程组解析。