We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.

在这项工作中，我们提出并分析了一种完全自适应的策略来解决带有随机数据的椭圆PDE。用于空间逼近的自适应网格细化的分层序列与随机空间中的自适应各向异性稀疏Smolyak网格相结合，以使计算成本最小化。我们策略的新颖之处在于，空间近似的层次取决于样本，因此可以单独优化每个搭配点的计算工作量。我们概述了自适应多级算法的收敛性和计算复杂性，并为每个级别的容错提供了最佳选择。