Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation
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DOI：
10.1137/10081157x
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发表时间：
2012-04
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影响因子：
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通讯作者：
M. Drohmann;B. Haasdonk;Mario Ohlberger
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文献类型：
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作者：
M. Drohmann;B. Haasdonk;Mario Ohlberger
本文提出了一种新的方法来处理参数化发展方程的约化基近似中的非线性算子。我们的方法是基于经验插值的非线性微分算子及其Frechet导数。有效的离线/在线分解获得离散算子，允许一个有效的评价为一定的插值泛函。推导了相应的缩减基方法的后验误差估计，并进行了数值分析。我们介绍了一种新的算法，PODEI贪婪算法，它构造的经验插值和数值方案的同步方式减少基空间。该方法适用于非线性抛物型和双曲型方程的显式或隐式有限体积离散。我们表明，由此产生的减少计划是能够捕捉到光滑和不连续的解决方案的演变。在问题对称的情况下，该方法实现了自动和直观的空间压缩甚至空间降维。我们进行实证调查的误差收敛和运行时间。在所有情况下，我们获得了良好的运行时加速。
We present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Frechet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEI-greedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and run-times. In all cases we obtain a good run-time acceleration.