Desbrun, Mathieu and Donaldson, Roger D. and Owhadi, Houman (2009) Discrete Geometric Structures in Homogenization and Inverse Homogenization with Application to EIT. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechAUTHORS:20111011163848887

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Abstract
We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x) in dimension two. We show that conductivity coefficients are in onetoone correspondence with divergencefree matrices and convex functions s(x) over the domain Ω. Although homogenization is a nonlinear and noninjective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of Ω when reexpressed using convex functions, and is a volume averaging operator when reexpressed with divergencefree matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain an optimally robust homogenization algorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy to weighted Delaunay triangulations. Next, we consider inverse homogenization, that is, the recovery of the microstructure from macroscopic information, a problem which is known to be both nonlinear and severly illposed. We show how to decompose this reconstruction into a linear illposed problem and a wellposed nonlinear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT) in dimension two. It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary DirichlettoNeumann map. This is of practical importance since the EIT problem always admits a unique solution in the space of divergencefree matrices and is stable with respect to Gconvergence in that space (this property fails for isotropic matrices). As such, we suggest that the space of convex functions is the natural space to use to parameterize solutions of the EIT problem.
Item Type:  Report or Paper (Technical Report)  

Group:  Applied & Computational Mathematics  
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Record Number:  CaltechAUTHORS:20111011163848887  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20111011163848887  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  27172  
Collection:  CaltechACMTR  
Deposited By:  Kristin Buxton  
Deposited On:  19 Oct 2011 18:16  
Last Modified:  26 Dec 2012 14:15 
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