Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

Abstract

Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical micro-structures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macro-scales we observe and the underlying micro-scale models we assume. Among the various applications benefiting from homogenization, Electric Impedance Tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardio-pulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images.In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform elliptic operators with rough conductivity coefficients in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions over the domain. 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 the domain when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain a 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. We then consider inverse homogenization, which is known to be both non-linear and severely illposed, but that we can decompose into a linear ill-posed problem and a well-posed non-linear problem. Finally, our new geometric approach to homogenization and inverse homogenization is applied to EIT.

On the equilibrium of simplicial masonry structures

“…Our discretization provides a formal backdrop to the equations used in thrust network analysis [Block and Ochsendorf 2007], and inherits the convergence and accuracy analysis tools available in the finite elements literature [Desbrun et al 2013]. Moreover, we managed to completely characterize the space of valid discrete equilibrium solutions, which will allow us to solve for self-supporting structures using the reduced set of variables w and c that fully describe the Figure 2: Effect of harmonic one-forms.…”

“…(5)). This exact case (up to sign) appeared in the very different context of electric impedance tomography with rough conductivity coefficients in [Desbrun et al 2013], where a numerical homogenization (or upscaling) of the equations were proposed via a harmonic change of coordinates. In particular, it was pointed out that a symmetric tensor σ in 2D can be conveniently, but rigorously discretized on a simplicial mesh T by a scalar value σij for every (unoriented) edge ij using the piecewise linear basis functions φi and φj as:…”

“…In the discrete setting, this function corresponds to the integrable part of the orthogonal dual mesh associated to the discrete stress values σij and thus it is defined through the zero-form w via ψ(u) = 1 2 i ui 2 − wi φi(u). Geometrically, the Airy function can be seen as the lifting of the 2D mesh T to a paraboloid of height ψ(u) as used in [Desbrun et al 2013]; see Fig. 1.…”

On the Correspondence between 2D Force Networks and Polyhedral Stress Functions

“…semi-axes of the elliptic basis. On considering the stress function defined by (15) we generate the following Pucher stresses over the membrane platform (16) Let us assume q = 1, a = 11.26, b = 5.63, h = 10 (in abstract units). As in the previous example, we study a structured and an unstructured truss models of the problem under examination.…”

Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients

“…Since (see [62]) Gamblets are also natural basis functions for numerical homogenization [104,3,46,30,70,31,9,2,25,90,102,72,51,73,45,76] they can also be employed to achieve sub-linear complexity under sufficient regularity of source terms and initial conditions (see [71,69,72] and Remark 4.3).…”