Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.
翻译:随机参数PDE的数值方法可大大受益于适应性改进办法,特别是当在随机性Galerkin和随机性合用法中计算功能近似值时。这项工作涉及适应性Galerkin FEM的非侵入性概括法和剩余误差估计法。它将随机性最低方位方法的非侵入性与对随机性Galerkin方法的事后误差分析相结合。拟议方法使用变异性蒙特卡洛法以高效的等级振标格式获得加勒金投影的准最佳低级近近似法。我们用可靠的误差估计法指导一种适应性改进算法。相对于随机性加勒金方法,这种方法很容易适用于广泛的问题,能够完全自动调整所有离散性参数。基准示例包括缩缩和(约束性)对数参数字段,以显示非侵入性适应性算法的性能,显示在类内的最佳性能。