Matrix valued inverse problems on graphs with application to elastodynamic networks

TL;DR

This paper studies the inverse problem of determining matrix-valued parameters in graph-based systems from boundary measurements, with applications to elastodynamic networks, providing conditions for uniqueness, explicit formulas, and numerical implications.

Contribution

It generalizes resistor network inverse problems to matrix-valued weights, establishes conditions for uniqueness, and derives explicit Jacobian formulas with applications to elastodynamic networks.

Findings

Unique solvability conditions for matrix-valued inverse problems.

Explicit Jacobian formulas for parameter-to-data mapping.

Applicability to elastodynamic networks with complex weights.

Abstract

We consider the inverse problem of finding matrix valued edge or nodal 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 elastodynamic networks (networks of springs, masses and dampers) is presented.