Stochastic Subspace via Probabilistic Principal Component Analysis for Characterizing Model Error

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.

Published in Computational Mechanics, a journal

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