Reduced-basis output bound methods for parabolic problems
Reduced-basis output bound methods for parabolic problems
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DOI：
10.1093/imanum/dri044
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发表时间：
2006-07
影响因子：
2.1
通讯作者：
D. Rovas;L. Machiels;Y. Maday
中科院分区：
文献类型：
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作者：
D. Rovas;L. Machiels;Y. Maday
在本文中，我们将早期针对椭圆问题开发的降基输出界限方法扩展到“参数化抛物线”偏微分方程描述的问题。本文的基本新成分和新颖性在于问题的提出和解决中存在时间。首先，在不假设时间离散化的情况下，提出了一个简化基过程来“有效地”计算抛物线问题的解和感兴趣的“相关”输出的精确近似值。此外，我们开发了一个误差估计程序来“事后验证”我们输出预测的准确性。其次，使用不连续伽辽金法进行时间离散化，分析了半离散情况的减基法和输出有界过程。在这两种情况下，缩减基都是通过在时间和参数上拍摄解的“快照”来构建的：从这个意义上说，该方法接近本征正交分解（POD）。
In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by ‘parameterized parabolic’ partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest. In addition, we develop an error estimation procedure to ‘a posteriori validate’ the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analysed for the semi-discrete case. In both cases the reduced-basis is constructed by taking ‘snapshots’ of the solution both in time and in the parameters: in that sense the method is close to Proper Orthogonal Decomposition (POD).