We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic partial differential equation, where the diffusion coefficient is parametrized by means of a Karhunen-Loève 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-Loève 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.

中文翻译：

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不确定性量化的维度自适应组合技术

我们提出了一种自适应算法，用于计算感兴趣的量，涉及随机椭圆偏微分方程的解，其中扩散系数通过 Karhunen-Loève 展开进行参数化。等效参数问题的逼近需要将可数无限维参数空间限制为有限维参数集、空间离散化以及参数变量的逼近。我们考虑在这些近似方向之间采用稀疏网格方法，以减少计算量，并提出尺寸自适应组合技术。此外，采用稀疏网格求积法进行高维参数逼近，并同时与空间和随机逼近进行平衡。我们的自适应算法基于效益成本比构建稀疏网格近似，从而预先不需要 Karhunen-Loève 系数的规律性和衰减。当算法根据参数变量的各向异性进行调整时，会检测并利用衰减。我们提供了具有对数正态渗透率场的达西问题的数值示例，这说明了该算法的良好性能。对于足够平滑的随机场，我们本质上将收敛的空间顺序恢复为相对于计算成本的渐近收敛率。