Matrix valued inverse problems on graphs with application to mass-spring-damper systems

Travis G. Draper; Fernando Guevara Vasquez; Justin Cheuk-Lum Tse; Toren E. Wallengren; Kenneth Zheng

doi:10.3934/nhm.2020001

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2020, Volume 15, Issue 1: 1-28. Doi: 10.3934/nhm.2020001

This issue Previous Article Next Article Mathematical analysis of transmission properties of electromagnetic meta-materials

Matrix valued inverse problems on graphs with application to mass-spring-damper systems

Mathematics Department, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA

^* Corresponding author: F. Guevara Vasquez
^* Corresponding author: F. Guevara Vasquez

Received: May 31, 2018

Revised: July 31, 2019

Published: December 2019

This work was supported by the National Science Foundation grants DMS-1411577 and DMS-1439786.

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Abstract

We consider the inverse problem of finding matrix valued edge or node quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and current measurements at a few nodes, but where the voltages and currents are vector valued. The measurements come from solving a series of Dirichlet problems, i.e. finding vector valued voltages at some interior nodes from voltages prescribed at the boundary nodes. We give conditions under which the Dirichlet problem admits a unique solution and study the degenerate case where the edge weights are rank deficient. Under mild conditions, the map that associates the matrix valued parameters to boundary data is analytic. This has practical consequences to iterative methods for solving the inverse problem numerically and to local uniqueness of the inverse problem. Our results allow for complex valued weights and give also explicit formulas for the Jacobian of the parameter to data map in terms of certain products of Dirichlet problem solutions. An application to inverse problems arising in networks of springs, masses and dampers is presented.

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Mathematics Subject Classification: Primary: 05C22, 05C50; Secondary: 35R30.

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Figure 1. A simple spring, mass and damper network that we use in example 1. Here $ V = \{0, 1, 2, 3\} $ and $ E = \{\{0, 1\}, \{0, 2\}, \{0, 3\}\} $. The node positions at equilibrium are given in black and correspond to $ x_0 = (0, 0)^T $, and $ x_i = (\cos(2\pi i / 3), \sin(2\pi i / 3))^T $, $ i = 1, \ldots, 3 $. A displaced configuration is given in blue, where the new node positions are $ x_i + u_i $, with displacements $ u_i $ for $ i = 0, \ldots, 3 $

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Figure 2. The Laplacian for a scalar conductivity on the cylindrical graph $ C \equiv P_5 \times P_3 $ can be seen as a matrix valued Schrödinger operator on the graph $ P_5 $, as explained in example 2. To fix ideas, $ s^4 \in \mathbb{R}^2 $ represents the conductivities of $ C $ within the $ 4- $th group in red and defines the matrix valued Schrödinger potential $ q(4) $. The conductivity $ s^{2, 3} \in \mathbb{R}^3 $ represents the conductivities of the 3 edges between the 2nd and 3rd group and is used to define the matrix valued conductivity $ \sigma(\{2, 3\}) $

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