This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.

中文翻译：

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用于参数化问题的适当广义分解近似的随机残差误差估计器

本文介绍了一种新的误差估计器，用于参数化方程的适当广义分解 (PGD) 近似。估计量本质上是随机的：它建立在高斯图的浓度不等式和随机右侧的伴随问题之上，我们使用 PGD 对其进行近似。这种随机误差估计器的有效性可以任意接近于 1 的概率，允许估计与任何用户定义的范数有关的误差以及某些感兴趣数量的误差。证明了误差估计器的性能，并与一些现有的用于参数化时谐弹性动力学问题的 PGD 误差估计器和具有高维参数空间的线性弹性参数化方程进行了比较。