The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

翻译：这项工作的主题是用于二级椭圆部分差异方程式的一种新的随机伸缩加热金法,配以随机扩散系数。它把随机变数中的操作员压缩与空间变量中的树基样条纹波子近似值结合起来。根据特定随机扩散系数的多层扩展,该方法可以实现最佳计算复杂性,直至对数系数。与现有结果相反,当可实现的趋同率受随机字段的规律性而不是空间近似顺序的限制时,这尤其有效。聚合和复杂估计值通过数字实验加以说明。