Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis
Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis
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DOI：
10.1002/nme.2540
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发表时间：
2009-11-19
影响因子：
2.9
通讯作者：
Nair, P. B.
中科院分区：
文献类型：
--
作者：
Audouze, C.;De Vuyst, F.;Nair, P. B.
本文提出了一种构造参数化稳态偏微分方程有限元/有限体积离散解的低阶代理模型的方法。在物理空间和参数空间中构造适当的正交分解模式，使我们能够仅使用少量系数来表示高维离散解。针对高维参数空间的问题，提出了一种增量贪婪算法。对于数值实验和验证，几个非线性稳态对流扩散反应问题被认为是：首先在一个空间维度的两个参数，然后在两个空间维度的两个和五个参数。在具有两个参数的二维空间情况下，示出了7 x 7系数矩阵足以精确地再现预期解，而在五个参数问题中，示出了13 x 6系数矩阵以足够的精度再现解。所提出的方法有望在参数变化研究、不确定性分析、反问题和优化设计中得到应用。版权所有（c）2009约翰威利父子有限公司。
This paper presents a methodology for constructing low-order surrogate models of finite element/finite volume discrete solutions of parameterized steady-state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high-dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high-dimensional parameter spaces. For numerical experiments and validation, several non-linear steady-state convection-diffusion-reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two-dimensional spatial case with two parameters, it is shown that a 7 x 7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13 x 6 coefficient matrix is shown to reproduce the Solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation Studies, uncertainty analysis, inverse problems and optimal design. Copyright (c) 2009 John Wiley & Sons, Ltd.