一种用于具有局部不确定性和非线性的半线性椭圆方程的多尺度方法
A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities
AI总结：
本文提出了一种多尺度数值方法，用于求解具有局部不确定性和非线性的半线性椭圆随机偏微分方程，通过域分解方法将不同不确定性和非线性源包含在感兴趣的子域（称为补丁）中，并通过迭代算法求解一系列线性全局问题和非线性局部问题，利用自适应采样基于最小二乘法构建稀疏多项式近似来解决非线性局部问题，证明了算法的一致性、收敛性和鲁棒性，并通过数值实验展示了方法的性能。
摘要：
本文提出了一种多尺度数值方法，用于求解具有局部不确定性和非线性的半线性椭圆随机偏微分方程，其中不确定性由一组随机参数建模。该方法依赖于域分解方法，引入了若干感兴趣的子域（称为补丁），包含不同的不确定性和非线性源。然后引入了一种迭代算法，要求求解一系列具有确定性算子和不确定右端项的线性全局问题，以及在补丁上具有不确定算子和/或右端项的非线性局部问题。非线性局部问题通过自适应采样基于最小二乘法的方法来构建局部解关于随机参数的稀疏多项式近似。在一般半线性椭圆算子假设下证明了算法的一致性、收敛性和鲁棒性。还引入了一种收敛加速技术（Aitken's动态松弛）以加速算法的收敛。通过在稳态非线性扩散-反应问题上的数值实验展示了所提方法的性能。
英文摘要：
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken's dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.
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