The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.
翻译:本文考虑了一组参数等离子体部分差异方程(PDEs),其中系数和右侧函数取决于无限多(不确定)参数。我们引入了两级后端估计器,以控制多层随机焦差加热金近似值的能源错误。我们证明两级估计器总是为未知近似误差提供较低的界限,而上界则相当于饱和性假设。我们提出并用经验比较了三种适应性算法,在这种算法中,估计器的结构被用来进行空间改进和参数浓缩。文件还探讨了计算多层随机加热近似值的落实问题。