论文标题
椭圆形部分微分方程中的次级高斯随机字段
Subordinated Gaussian Random Fields in Elliptic Partial Differential Equations
论文作者
论文摘要
为了在不确定的异质\断裂介质中对地下流进行建模，具有不连续的随机扩散系数（也称为随机场）的椭圆形方程。如果有一维参数空间，Lévy过程可以跳跃并在所使用的分布中显示出极大的灵活性。但是，在各种情况下（例如微结构建模），一维参数空间还不够。莱维过程的经典扩展在两个参数维度上遭受了这样一个事实，即它们不允许空间不连续性。在本文中，采用了一种新的从属方法来在二维空间参数域上产生lévy型不连续的随机场。证明了对一般椭圆部分微分方程的（路径）解决方案的存在和唯一性，并证明了扩散系数和提供的相应解决方案的近似理论。此外，使用蒙特卡洛方法对有限元离散化的数值示例验证了我们的理论结果。
To model subsurface flow in uncertain heterogeneous\ fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities. In this paper a new subordination approach is employed to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.
