具有微扰缩回的动态低秩逼近的动态正交龙格-库塔格式

A. Charous, Pierre FJ Lermusiaux
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引用次数: 5

摘要

。无论是由于计算域的绝对大小,所需的精细分辨率,还是动力学的多重尺度和随机性,系统的维数必须经常降低,以便感兴趣的问题在计算上易于处理。在本文中,我们开发了7个时间积分方案,有效和准确地演化了一个8系统的低秩近似的动力学。利用微分几何的方法,分析了固定秩矩阵流形的高阶曲率在每个时间步所产生的误差。我们首先得到一个新颖的,显式的,计算成本低廉的算法集,我们称之为微扰缩回,并表明该集收敛于一个理想的缩回,通过减少我们定义的投影-缩回误差,该理想缩回最优且精确地投影到固定秩矩阵的流形上。13此外,每个微扰回缩本身表现出高阶收敛性,达到全阶解的最佳低- 14阶近似。利用微扰缩回,我们开发了一类新的积分技术,我们称之为动态正交龙格-库塔(DORK) 16格式。DORK方案沿着非线性流形集成,更新子空间,我们在其上投影系统的动力学,因为它是集成的。通过数值测试用例,我们展示了矩阵加法、实时数据压缩以及确定性和随机偏微分方程的方案。我们发现DORK方案通过结合动态非线性流形的高阶曲率的知识而具有很高的准确性,并且通过限制表示演化动力学所需的增长秩而具有很高的计算效率。在时间步长期间,有22种方案可以解释降阶积分空间的774次演化。我们表明,775 DORK方案(i)通过结合动态知识而高度精确,776非线性流形从阶段到阶段的高阶曲率,以及(ii)通过限制表示演化动力学L所需的增长秩来计算777效率。在我们的实时数据压缩示例中,我们表明,当动力学是数据驱动的(而不是模型驱动的)时,可以使用这些缩回,并且具有高阶修正的摄动缩回以较低的速度从最佳低秩近似漂移。注意,调整动力学计算的方式以纠正漂移,这对于获得最准确的方案非常重要。最后,我们证明了在确定性和随机微分方程中可以使用缩回:动态低秩785近似和缩回与我们选择压缩的数学786空间的性质无关,无论是确定性的还是随机的。在这两种情况下,787低秩近似不仅快速收敛到最佳近似,而且收敛到全秩解788。在随机情况下,这为非高斯统计非线性问题(例如,790[42,44,41,40])中的789不确定性量化提供了一种有效的方法。791
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Dynamically Orthogonal Runge-Kutta Schemes with Perturbative Retractions for the Dynamical Low-Rank Approximation
. Whether due to the sheer size of a computational domain, the fine resolution required, 5 or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often 6 be reduced so that problems of interest become computationally tractable. In this paper, we develop 7 retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a 8 system’s low-rank approximation. Through differential geometry, we analyze the error incurred 9 at each time step due to the high-order curvature of the manifold of fixed-rank matrices. We first 10 obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative 11 retractions and show that the set converges to an ideal retraction that projects optimally and exactly 12 to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. 13 Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low- 14 rank approximation of the full-rank solution. Using perturbative retractions, we then develop a new 15 class of integration techniques that we refer to as dynamically orthogonal Runge-Kutta (DORK) 16 schemes. DORK schemes integrate along the nonlinear manifold, updating the subspace upon which 17 we project the system’s dynamics as it is integrated. Through numerical test cases, we demonstrate 18 our schemes for matrix addition, real-time data compression, and deterministic and stochastic partial 19 differential equations. We find that DORK schemes are highly accurate by incorporating knowledge 20 of the dynamic, nonlinear manifold’s high-order curvature and computationally efficient by limiting 21 the growing rank needed to represent the evolving dynamics. 22 schemes that account for the 774 evolution of the reduced-order integration space during the time-step. We show that 775 DORK schemes are (i) highly accurate by incorporating knowledge of the dynamic, 776 nonlinear manifold’s high-order curvature from stage to stage and (ii) computationally 777 efficient by limiting the growing rank needed to represent the evolving dynamics L . In 778 our real-time data compression example, we show that these retractions may be used 779 when the dynamics are data-driven (rather than model-driven), and the perturbative 780 retractions with high-order corrections drift from the best low-rank approximation at 781 a much slower rate than retractions with low-order corrections. Note that adjusting 782 how the dynamics are calculated to correct the drift in the first place is important 783 to obtain the most accurate scheme. Lastly, we show that the retractions may be 784 used in deterministic and stochastic differential equations: the dynamical low-rank 785 approximation and the retractions are agnostic to the nature of the mathematical 786 spaces we choose to compress, whether deterministic or stochastic. In both cases, the 787 low-rank approximation converges quickly not just to the best approximation, but to 788 the full-rank solution. In the stochastic case, this offers an efficient methodology for 789 uncertainty quantification in nonlinear problems with non-Gaussian statistics (e.g., 790 [42, 44, 41, 40]). 791
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