NONLINEAR MODEL REDUCTION VIA DISCRETE EMPIRICAL INTERPOLATION
NONLINEAR MODEL REDUCTION VIA DISCRETE EMPIRICAL INTERPOLATION
复制标题
DOI：
10.1137/090766498
复制
发表时间：
2010-01-01
影响因子：
3.1
通讯作者：
Sorensen, Danny C.
中科院分区：
文献类型：
--
作者：
Chaturantabut, Saifon;Sorensen, Danny C.
提出了一种称为离散经验插值法的降维方法，该方法大大降低了用于建立含时和/或参数非线性偏微分方程组(PDE)降阶模型的常用的本征正交分解(POD)方法的计算复杂度。在一般非线性存在的情况下，标准的POD-Galerkin技术降维，因为存在的变量要少得多，但计算非线性项的复杂性仍然是原始问题的复杂性。原始经验插值法(EIM)是对POD的一种改进，它将简化模型的非线性项的计算复杂度降低到与POD得到的简化变量个数成正比的代价。提出了一种离散经验插值法(DEIM)，这是一种适用于某类常微分方程组降维的变种。它适用于含时偏微分方程组的有限差分离散和/或含参数定态问题的常微分方程组。然而，这种方法只需稍加修改就可以推广到任意的非线性常微分方程组。我们的贡献是在有限维环境下极大地简化了对EIM的描述，该环境对逼近质量具有误差界。将Deim方法应用于一维Fitzhugh-Nagumo方程的有限差分离散，在完全捕捉非线性极限环行为的长时间积分中，将维度从1024降至5阶，误差可忽略不计。在状态空间降维和精度相近的情况下，我们还证明了该算法在更高的空间维度上的适用性。
A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.