SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 414-450, January 2020.
A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. Assuming a known micro structure, our aim is to recover a macroscopic scalar parameterization of the microscale tensor. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings.

中文翻译：

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椭圆多尺度逆问题的贝叶斯数值均化方法

SIAM / ASA不确定性量化期刊，第8卷，第1期，第414-450页，2020年1月。
介绍了一种基于数值均化和贝叶斯技术的多尺度逆问题求解新策略。我们考虑一类在微观尺度上变化的椭圆问题，我们的目标是使用基于数值均化和模型阶约的粗糙模型，从边界处精细尺度解决方案的测量结果中恢复高振荡张量。假定已知的微观结构，我们的目标是恢复微观尺度张量的宏观标量参数化。考虑到选择先行措施的各种可能性，我们对问题进行了严格的贝叶斯表述。我们证明了有效后验测度的正当性，并且通过G收敛，我们建立了有效后验测度与精细模型之间的联系。