# On 2D Inverse Problems/The inverse problems

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 is determined by its eigenvalues.