We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on coarse measurements. The approach is motivated by quasi-local numerical effective forward models that are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that an identification of the matrix representation related to these effective models is possible. On the one hand, this provides a reasonable surrogate in cases where a direct reconstruction is unfeasible due to a mismatch between the coarse data scale and the microscopic quantities to be reconstructed. On the other hand, the approach allows us to investigate the requirement for a certain non-locality in the context of numerical homogenization. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

我们考虑了基于粗糙测量为强异质介质中原型扩散过程构建有效模型的反问题。该方法是由准局部数值有效正向模型驱动的，该模型在周期性假设和标度分离之外证明是可靠的。这项工作的目的是表明可以识别与这些有效模型有关的矩阵表示。一方面，在由于粗略数据规模与要重建的微观量之间的不匹配而无法进行直接重建的情况下，这提供了合理的替代方案。另一方面，该方法允许我们在数值均质化的背景下研究对某些非局部性的要求。