Abstract
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.
Highlights
This paper focuses on the computational solution of multiscale inverse problems, i.e., where the quantities to be sought are related to an unknown microscopic mathematical model, while measurement data are available only on a much coarser scale
Compared to the locality of classical homogenization methods, numerical homogenization methods typically involve a slight deviation from local communication between the degrees of freedom which, in turn, leads to somewhat increased sparsity patterns of the corresponding system matrices
Our findings deviate from the numerical results in [13] which indicate that truly local numerical homogenization might always be possible
Summary
This paper focuses on the computational solution of multiscale inverse problems, i.e., where the quantities to be sought are related to an unknown microscopic mathematical model, while measurement data are available only on a much coarser scale. Effective models are the key to bridge the discrepancy between a microscopic coefficient and a coarse scale of interest in the forward setting They provide models that compute reliable approximations of the solution of a PDE even in the presence of microscopic quantities. If structural assumptions such as (local) periodicity or scale separation hold, classical homogenization methods (see, e.g., [21,28,42,43]) based on analytical homogenization theory can be used. Compared to the locality of classical homogenization methods, numerical homogenization methods typically involve a slight deviation from local communication between the degrees of freedom which, in turn, leads to somewhat increased sparsity patterns of the corresponding system matrices. Since this non-locality can be controlled, we refer to these methods as quasi-local
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