The inverse problems, that are the main objects of this book, have its roots in the study of the relationships between local and global properties of graphs embedded in surfaces (2D manifolds). The two-dimensional setting has an important structure, due to the duality between harmonic conjugate functions on the embedded graphs and the connections to classical special matrices, that can be viewed as two-dimensional objects. Also, the dimensions of the unknown parameters fit the dimension of the measurement data in several important instances of the inverse problems. The ideas of the solutions of the inverse problems are presented in a way bridging the discrete and continuous domain settings.

The techniques of solutions are motivated by the following commutative diagram:

The context of the inverse problems provides a unified point of view on the work of many great mathematicians, and even though there are references to many areas of mathematics in this book, it is practically self-contained, and is intended for a wide audience of readers interested in the subject. There are exercises, ranked by difficulty, through most of the chapters of the book and it can be used as a textbook for an applied mathematics class.