Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation
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Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation Oleg Balabanov1,2 · Anthony Nouy1 Received: 15 March 2018 / Accepted: 9 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the spaces of associated residuals. This approach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in lowdimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems. Keywords Model reduction · Reduced basis · Proper orthogonal decomposition · Random sketching · Subspace embedding Mathematics Subject Classification (2010) 15B52 · 35B30 · 65F99 · 65N15 Communicated by: Anthony Patera Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10444-019-09725-6) contains supplementary material, which is available to authorized users. Anthony Nouy
[email protected]
Extended author information available on the last page of the article.
O. Balabanov, A. Nouy
1 Introduction Projection-based model order reduction (MOR) methods, including the reduced basis (RB) method or proper orthogonal decomposition (POD), are popular approaches for approximating large-scale parameter-dependent equations (see the recent surveys and monographs [8, 9, 23, 31]). They can be considered in the contexts of optimization, uncertainty quantification, inverse problems, real-time simulations, etc. An essential feature of MOR methods is offline/online splitting of the computations. The construction of the reduced order (or surrogate) model, which is usually the most computationally demanding task, is performed during the offline stage. This stage consists of (i) the generation of a reduced approximation space with a greedy algorithm for RB method or a principal component analysis of
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