The inverse problems that we discuss in this book are the problems of inferring information about a graph or a manifold from the measurement data about solutions of difference and differential systems of equations defined on the domain of interest. Historically the inverse problems are split into two categories: inverse boundary problems and inverse spectral problems. In this book we will consider both and also the relationship between continuous inverse problems on manifolds and discrete inverse problems on the embedded graphs.
The inverse boundary problems are concerned w/ finding the local and global properties of graphs and manifolds from the boundary data of the solutions of difference and differential equations defined on them. One looks for properties of a domain from the spectral data of difference or differential operators defined on it.
Exercise (*). Show that the problem of finding roots of a polynomial can be restated as an inverse problem on the graph of the following type:
(Hint). The boundary data consists of values of the elementary symmetric functions of the weights a,b,c,d.
Exercise (*). Prove that the matrix of hitting probabilities of a rotation invariant planar graph is a circulant matrix, and it is determined by its eigenvalues.