基底适应与域分解在具有随机系数的稳态偏微分方程中的应用
Basis adaptation and domain decomposition for steady partial differential equations with random coefficients
AI总结：
本文提出了一种结合空间域分解与基底适应的新方法，用于求解高维随机参数空间下的稳态随机偏微分方程，通过局部准确解的拼接实现全局解的高效求解。
摘要：
我们提出了一种新的方法，用于求解具有高维随机参数空间的稳态随机偏微分方程。所提出的方法结合了空间域分解与每个子域中的基底适应。基底适应用于通过在每个子域中构建准确的低维表示来解决维度灾难问题，该表示包括随机偏微分方程解的概率密度函数及其主导统计矩。将基底适应限制在特定子域中，可以找到局部准确的解。然后将所有子域的解拼接在一起，以提供全局解。我们通过一个具有随机空间依赖系数的稳态扩散方程的数值实验来支持我们的构造。我们的结果表明，可以以显著减少的计算成本获得高精度的全局解。
英文摘要：
We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numerical experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.
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