Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1565-1594, January 2020.
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows us to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analyzed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modeling error and computational cost.

中文翻译：

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集成卡尔曼滤波器的多尺度逆问题

多尺度建模与仿真，第18卷，第4期，第1565-1594页，2020年1月。
我们提出了一种基于集合卡尔曼滤波器的新颖算法来解决涉及多尺度椭圆偏微分方程的反问题。我们的方法基于数​​值均化和有限元离散化，并允许我们以计算上不昂贵的方式从多尺度解的测量结果中恢复高度振荡的张量。针对多尺度和离散化参数分析了近似解的性质，并证明了收敛结果。提供了从贝叶斯角度对解的重新解释，并证明了有关Wasserstein距离的近似条件后验分布的收敛性。数值实验验证了我们的方法，特别强调了建模误差和计算成本。