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

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.

Networks with complex weights: Green function and power series

“…For the Markov chain with transition matrix P starting at vertex x, we have that G P V ˝px, yq is the expected number of visits in y before leaving the interior V ˝. Furthermore, it follows from [18] and [11] that also for complex weights with positive real part, I V ˝´P s|V ˝is invertible for every s P H r . See in particular the proof of [18,Theorem 2].…”

“…Our interest is in P " P s and the associated Dirichlet problem with complex weights. By [18] and [11] this problem has a unique solution v " v a whenever Re s ą 0. Indeed, the function v | B a V provides the (augmented) boundary data, and the solution can be given in two ways:…”

“…(Indeed, the boundary should rather be thought of as BV Y t8u.) We can again consider the power series (11). When is BV non-empty, we get that ( 14) Furthermore, the radius of convergence of the power series G P px, y|zq is 1{λpP q, and at z " 1{λpP q the latter either converges for all x, y or diverges for all x, y.…”

Networks with Complex Weights: Green Function and Power Series

“…For the Markov chain with transition matrix P starting at vertex x, we have that G P V ˝px, yq is the expected number of visits in y before leaving the interior V ˝. Furthermore, it follows from [21,22] that also for complex weights with positive real part, I V ˝´P s|V ˝is invertible for every s P H r . See in particular the proof of [22] (Theorem 2).…”

“…Our interest is in P " P s and the associated Dirichlet problem with complex weights. By [21,22] this problem has a unique solution v " v a whenever Re s ą 0. Indeed, the function v | B a V provides the (augmented) boundary data, and the solution can be given in two ways:…”