Date of Award

5-2022

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Advisor

Aaron Welters

Second Advisor

Marcus Hohlmann

Third Advisor

Jian Du

Fourth Advisor

Gnana Bhaskar Tenali

Abstract

In this thesis, we consider a class of Z-problems and their associated effective operators on Hilbert spaces which arise in effective media theory, especially within the theory of composites. We provide a unified approach to obtaining solutions of the Z-problem, formulas for the effective operator in terms of generalized Schur complements, and their associated variational principles (e.g., the Dirichlet minimization principle), while allowing for relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems. The Hilbert space framework developed here is inspired by the methods of orthogonal projections and Hodge decompositions. However, we focus on finite-dimensional Hilbert spaces. Our theoretical development utilizes the theory of block operator matrices, the Moore-Penrose pseudoinverse (as a replacement for the inverse), the generalized Schur complement, and the generalized principal pivot transform. With this unified framework, we are able to recover the classical minimization principle for the Schur complement and derive its extension, as well as provide a new maximization principle for the generalized principal pivot transform. We also present several applications. First, we give an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph and relate the DtN map to the effective operator of an associated Z-problem. Second, we show how the classical electrical conductivity of an electrical network (on a finite linear graph) is essentially the effective operator of an associated Z-problem. Next, we consider periodic linear graphs and develop a discrete analog to the periodic conductivity equation and effective conductivity in the continuum. Finally, we conclude with a discussion of future work and open problems based on this thesis.

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