SPACE-TIME LEAST-SQUARES PETROV-GALERKIN PROJECTION FOR NONLINEAR MODEL REDUCTION
SPACE-TIME LEAST-SQUARES PETROV-GALERKIN PROJECTION FOR NONLINEAR MODEL REDUCTION
复制标题
DOI：
10.1137/17m1120531
复制
发表时间：
2019-01-01
影响因子：
3.1
通讯作者：
Carlberg, Kevin
中科院分区：
文献类型：
--
作者：
Choi, Youngsoo;Carlberg, Kevin
这项工作提出了一种用于非线性动力系统模型简化的时空最小二乘 Petrov-Galerkin (ST-LSPG) 投影方法。与首先在空间维度上应用（Petrov-）Galerkin 投影，然后应用时间积分来数值求解所得低维动力系统的典型非线性模型简化方法相比，所提出的方法同时在空间和时间上应用投影。为了实现这一目标，该方法首先引入低维时空试验子空间，该子空间可以通过计算状态快照数据的张量分解来获得。然后，该方法通过最小化加权 l(2) 范数中所有空间和时间的时间离散化后产生的残差，计算该时空试验子空间中的离散最优近似。该范数可以定义为能够及时降低复杂性（即超降低），从而导致时空搭配和 ST-LSPG 方法的时空高斯牛顿近似张量 (GNAT) 变体。相对于典型的基于空间投影的非线性模型简化方法（例如伽辽金投影和最小二乘 Petrov-Galerkin 投影），该方法的优点包括减少动力系统的空间和时间维度，以及通过最佳时空近似误差限制解误差的先验误差界限，并且其稳定性常数随时间增长较慢。对流体动力学模型问题进行的数值示例表明，该方法能够相对于基于空间投影的降阶模型节省数量级的计算量，而不会牺牲固定时空离散化的精度。
This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted l(2) -norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.