We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen–Loève expansion entirely determines the regularity of the solution’s dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution’s mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.

中文翻译：

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具有各向异性各向异性扩散的椭圆PDE不确定度定量的多级方法

我们考虑具有随机各向异性扩散系数的椭圆扩散问题，其中，在随机矢量场给定的显着方向上，扩散强度不同于垂直于该显着方向的扩散强度。然后，Karhunen-Loève展开产生随机矢量场的参数化，因此也产生了椭圆扩散问题的解。我们证明，给定椭圆扩散问题的规则性，Karhunen-Loève展开的衰减完全决定了解决方案对随机参数的依赖性的规则性，同时考虑到这种较高的空间规则性。然后，该结果表明，在逼近感兴趣的量（例如解的均值或其第二矩）时，可以使用多级正交方法来降低计算复杂度，同时仍能达到预期的收敛速度。提供了三个空间维度的数值示例，以验证所提出的理论。