论文标题
部分可观测时空混沌系统的无模型预测
A Dimension-adaptive Combination Technique for Uncertainty Quantification
论文作者
论文摘要
我们提出了一种自适应算法，用于计算涉及随机椭圆PDE解决方案的利益量，其中扩散系数通过karhunen-loève膨胀进行了参数。等效参数问题的近似需要限制可观的无限二维参数空间到有限维参数集，空间离散化和参数变量的近似值。我们考虑了这些近似方向之间的稀疏网格方法，以减少计算工作并提出一种维度自适应组合技术。另外，采用了高维参数近似的稀疏网格正交正交正交正交，并与空间和随机近似同时平衡。我们的自适应算法基于利益成本的比率构建了稀疏的网格近似值，因此不需要事先需要规律性和Karhunen-Loève系数的衰减。当算法调整到参数变量中的各向异性时，检测并利用衰减。我们包括带有对数正态通透性场的Darcy问题的数值示例，其中说明了算法的良好性能：对于足够平滑的随机场，我们实际上是在相对于计算成本方面的渐近收敛速率恢复了收敛的空间顺序。
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-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.
