We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic PDE where the diffusion coefficient is parametrized by means of a Karhunen-Lo\`eve expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Lo\`eve coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm: For sufficiently smooth random fields, we essentially recover the rate of the spatial discretization as asymptotic convergence rate with respect to the computational cost.

翻译：我们提出了一种自适应算法，用于计算涉及随机椭圆偏微分方程解的感兴趣量，其中扩散系数通过Karhunen-Loève展开参数化。等效参数化问题的近似需要对可数无穷维参数空间进行有限维参数集限定、空间离散化以及参数变量的近似。我们采用这些近似方向之间的稀疏网格方法以减少计算负担，并提出了一种维度自适应组合技术。此外，我们使用稀疏网格求积进行高维参数近似，并同时平衡其与空间和随机近似的精度。我们的自适应算法基于效益-成本比构建稀疏网格近似，因此无需预先知道Karhunen-Loève系数的正则性及其衰减规律。该算法通过自适应调整参数变量的各向异性来检测并利用这一衰减规律。我们给出了对数正态渗透率场Darcy问题的数值算例，展示了该算法的良好性能：对于足够光滑的随机场，我们实质上恢复了空间离散化相对于计算成本的渐近收敛速率。