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 spatial order of convergence 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展开进行参数化。等价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截断、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来降低计算复杂度，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法进行高维参数近似，并同时平衡空间近似与随机近似。我们的自适应算法基于收益-成本比构造稀疏网格近似，因此无需预先假设Karhunen-Loève系数的正则性及其衰减规律。该算法通过自适应调整以检测并利用参数变量的各向异性衰减特征。我们以对数正态渗透率场的Darcy问题为例进行数值实验，结果表明该算法具有良好的性能：对于足够光滑的随机场，我们基本上恢复了以计算成本为参照的渐近收敛率所对应的空间收敛阶。