Volume 25, Issue 4
Generalized Multiscale Inversion for Heterogeneous Problems

Eric T. Chung, Yalchin Efendiev, Bangti Jin, Wing Tat Leung & Maria Vasilyeva

Commun. Comput. Phys., 25 (2019), pp. 1213-1234.

Published online: 2018-12

Preview Purchase PDF 98 2405
Export citation
  • Abstract

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid medium properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the scale of coarse grids. In these cases, grid refinement can lead to many degrees of freedom, and thus numerically unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach by numerical results for fractured media.

  • Keywords

Multiscale inversion, multiscale problem, generalized multiscale finite element method, coarse-grid.

  • AMS Subject Headings

65M32, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-25-1213, author = {}, title = {Generalized Multiscale Inversion for Heterogeneous Problems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {4}, pages = {1213--1234}, abstract = {

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid medium properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the scale of coarse grids. In these cases, grid refinement can lead to many degrees of freedom, and thus numerically unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach by numerical results for fractured media.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0184}, url = {http://global-sci.org/intro/article_detail/cicp/12896.html} }
TY - JOUR T1 - Generalized Multiscale Inversion for Heterogeneous Problems JO - Communications in Computational Physics VL - 4 SP - 1213 EP - 1234 PY - 2018 DA - 2018/12 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0184 UR - https://global-sci.org/intro/article_detail/cicp/12896.html KW - Multiscale inversion, multiscale problem, generalized multiscale finite element method, coarse-grid. AB -

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid medium properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the scale of coarse grids. In these cases, grid refinement can lead to many degrees of freedom, and thus numerically unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach by numerical results for fractured media.

Eric T. Chung, Yalchin Efendiev, Bangti Jin, Wing Tat Leung & Maria Vasilyeva. (2020). Generalized Multiscale Inversion for Heterogeneous Problems. Communications in Computational Physics. 25 (4). 1213-1234. doi:10.4208/cicp.OA-2017-0184
Copy to clipboard
The citation has been copied to your clipboard