Randomized Residual-Based Error Estimators for Parametrized Equations
Randomized Residual-Based Error Estimators for Parametrized Equations
复制标题
参数化方程的随机残差误差估计器
DOI：
10.1137/18m120364x
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
A. Patera
中科院分区：
文献类型：
--
作者：
K. Smetana;O. Zahm;A. Patera
提出了参数化（偏）微分方程降阶近似的随机后验误差估计。该误差估计具有几个重要的性质：在给定的高概率下，有效性接近于1，具有给定的上下界;不需要计算稳定性（误差或inf-sup）常数;评估后验误差估计器的在线成本与找到降阶近似的成本相称;概率界限扩展到许多查询，而成本仅适度增加。为了构建此估计器，我们首先使用高斯随机向量使用蒙特-卡罗估计器估计误差的范数，高斯随机向量的协方差根据所需的误差度量来选择，例如用户定义的范数或感兴趣的量。然后，我们引入了一个随机右手边的解决方案，它允许我们重写的误差估计的原始方程的残差方面的对偶问题。为了得到一个快速估计的估计量，可以使用模型降阶方法来近似随机对偶解。在这里，我们提出了一个贪婪的算法，这是由一个标量的兴趣取决于误差估计。多参数Helmholtz问题的数值实验表明，该策略产生相当低维的约化对偶空间。
We propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte-Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g. user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multi-parametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces.