[Paper Review] Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT

Mathieu Desbrun, Roger Donaldson+1 more|ArXiv.org|Apr 17, 2009

Numerical methods in inverse problems55 references3 citations

TL;DR

This paper introduces a geometric framework that re-expresses conductivity homogenization as a linear interpolation over convex functions or volume averaging over divergence-free matrices, enabling robust, optimal homogenization via weighted Delaunay triangulations. Applied to 2D Electrical Impedance Tomography (EIT), it enables stable, anisotropic reconstruction of conductivity from boundary data by parameterizing solutions in the space of convex functions, achieving previously unattained resolution of fine-scale microstructures.

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 one-to-one correspondence with divergence-free matrices and convex functions $s(x)$ over the domain $Ω$. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of $Ω$ when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). 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 Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to $G$-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem.

Motivation & Objective

• To develop a geometric framework for homogenization of elliptic operators with rough conductivity coefficients without relying on small-scale parameters.
• To address the ill-posed inverse homogenization problem in EIT by decomposing it into a linear ill-posed and a well-posed non-linear problem.
• To provide a stable, robust reconstruction method for EIT that parameterizes conductivity in the space of convex functions, improving resolution of anisotropic microstructures.
• To extend the optimality of Delaunay triangulations to weighted, Q-adapted meshes for discrete convex function interpolation.

Proposed method

• Parameterize conductivity σ(x) via a one-to-one correspondence with divergence-free matrices Q and convex functions s(x), enabling re-expression of homogenization as linear interpolation or volume averaging.
• Use optimal weighted Delaunay triangulations to interpolate convex functions s(x), ensuring global optimality in minimizing discrete Dirichlet energy.
• Construct a discrete Dirichlet-to-Neumann (DtN) map Λ_sh from s^h_i using quadratic interpolation on triangle stencils, including ghost vertices for boundary stability.
• Solve the inverse EIT problem via optimization: minimize data misfit plus total variation regularization of tr(Q^h), subject to q^h_ij ≥ 0.
• Compute the Jacobian of the data misfit term using a primal-adjoint method for efficient optimization with IpOpt.
• Use the trace of Q^h as a regularization term for computational tractability, with experimental tuning of Tikhonov parameter α.

Experimental results

Research questions

• RQ1 Can homogenization of rough conductivity coefficients be reformulated as a linear operator in a geometric space, avoiding non-linearities?
• RQ2 How can optimal meshes be constructed for discrete convex function interpolation to ensure stability and accuracy in homogenization?
• RQ3 Can the inverse EIT problem be decomposed into a well-posed non-linear problem and a linear ill-posed problem, enabling stable reconstruction?
• RQ4 Is parameterizing conductivity via convex functions more effective than standard parameterizations for resolving fine-scale anisotropic microstructures in EIT?
• RQ5 Can the framework reconstruct up-scaled representations of laminar microstructures below the stable resolution limit?

Key findings

• Homogenization becomes a linear interpolation operator when expressed in the space of convex functions s(x), and a volume averaging operator in the space of divergence-free matrices Q.
• Weighted Delaunay triangulations based on convex functions achieve global optimality in minimizing discrete Dirichlet energy, extending the properties of standard Delaunay triangulations.
• The method successfully reconstructs anisotropic conductivity patterns in EIT, including the orientation and strength of anisotropy, as visualized via eigenvalue differences of Q.
• The reconstruction of a laminated microstructure (Figure 2.2) achieves, to the authors' knowledge, a previously unrealized level of detail, capturing up-scaled anisotropy despite sub-resolution pitch.
• Regularization using tr(Q^h) outperforms det(Q^h) in computational efficiency without sacrificing reconstruction quality, as confirmed by numerical experiments.
• The method avoids underestimation of dynamic range in conductivity reconstructions, unlike some standard methods, and maintains stability under G-convergence in the divergence-free matrix space.