Title: Stochastic Subspace via Probabilistic Principal Component Analysis for Characterizing Model Error Show Chinese title

Title: 基于概率主成分分析的随机子空间用于表征模型误差

Abstract: This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space -- commonly known as a snapshot matrix -- this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to characterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We demonstrate the performance of the proposed method via several numerical examples in computational mechanics and structural dynamics. Show Chinese abstract

Abstract: 本文提出了一种基于概率主成分分析（PCA）的子空间概率模型。 给定嵌入空间中的向量样本——通常称为快照矩阵——该方法利用源自概率PCA的量来构造样本矩阵以及主要子空间的分布。 它适用于基于投影的降阶建模方法，如恰当正交分解及相关模型简化方法。 由此构建的随机子空间可以用于表征计算力学中的模型形式不确定性。 所提出的方法具有多个优点： (1) 它由概率PCA自然推导，并且对于由此诱导的随机矩阵模型具有解析形式；(2) 默认情况下，它可以满足线性约束，例如各种边界条件；(3) 它只有一个超参数，这显著简化了训练过程；(4) 其算法非常容易实现。 我们通过计算力学和结构动力学中的几个数值例子展示了所提出方法的性能。