斜投影法求解时变基非线性随机偏微分方程的可伸缩秩自适应降阶建模

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2023-10-01 DOI:10.1098/rspa.2023.0320
M. Donello, G. Palkar, M. H. Naderi, D. C. Del Rey Fernández, H. Babaee
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引用次数: 1

摘要

时间相关基降阶模型(TDB - ROMs)已成功地用于逼近非线性随机偏微分方程(PDEs)的解。对于许多实际问题,离散这些偏微分方程会导致大量的矩阵微分方程(MDEs),而使用传统方法求解这些方程过于昂贵。虽然TDB rom有可能显著减少这种计算负担,但它们仍然面临以下挑战:(i)一般非线性的效率低下,(ii)侵入性实现,(iii)存在小奇异值时的病态以及(iv)由于固定秩而导致的误差积累。为此,我们提出了一种可扩展的方法来解决TDB rom,该方法具有计算效率高、侵入性小、在存在小奇异值时具有鲁棒性、秩自适应和高度并行性。这些有利的特性是通过斜投影实现的,这需要在少量的行和列上评估MDE。使用离散经验插值方法(DEIM)选择列和行,该方法产生接近最优的矩阵低秩近似。我们证明了所提出的算法等价于一个CUR矩阵分解。数值结果证明了该方法对多种问题的准确性、有效性和鲁棒性。
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Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases
Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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