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Type of Document	Dissertation
Author	Donaldson, Roger David
Author's Email Address	rdonald@acm.caltech.edu
URN	etd-05212008-164705
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-05212008-164705
Title	Discrete geometric homogenisation and inverse homogenisation of an elliptic operator
Degree	PhD
Option	Applied and Computational Mathematics
Advisory Committee	Advisor Name Title Houman Owhadi Committee Chair Jerrold Marsden Committee Member Mathieu Desbrun Committee Member Peter Schroeder Committee Member Thomas Hou Committee Member
Keywords	FEM upscaling metric-based up-scaling inverse problems discrete differential geometry electric impedance tomography conductivity homogenization finite element method metric convexity variational mesh generation anisotropy anisotropic weighted Delaunay triangulation Dirichlet Neumann map multiscale multi-scale
Date of Defense	2008-05-02
Availability	mixed
Abstract We show how to parameterise a homogenised conductivity in $R^2$ by a scalar function $s(x)$, despite the fact that the conductivity parameter in the related up-scaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of $s(x)$, and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated domains. We apply the parameterisation in three contexts: (i) sampling $s(x)$ produces a family of stiffness matrices representing the elliptic operator over a hierarchy of scales; (ii) the curvature of $s(x)$ directs the construction of meshes well-adapted to the anisotropy of the operator, improving the conditioning of the stiffness matrix and interpolation properties of the mesh; and (iii) using electric impedance tomography to reconstruct $s(x)$ recovers the up-scaled conductivity, which while anisotropic, is unique. Extensions of the parameterisation to $R^3$ are introduced.
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