User:DVD206/On 2D Inverse Problems

On Inverse Problems in 2D

Dedicated to Nicole DeLaittre

Summary

 * The main object of study of this book is the relationship between local and global properties of two-dimensional manifolds (surfaces) and embedded graphs. The dimension of the unknown parameter fits the dimension of the data of the measurements in several important instances of the inverse problems. Also, two-dimensional setting has an additional structure, due to the duality between harmonic functions on embedded graphs and manifolds and the connection to special matrices. The context of the inverse problems provides a unified point of view on the work of many great mathematicians. Some of the problems simplify significantly in the graph theoretical setting, but their solutions nevertheless convey the main ideas of the solutions for their continuous analogs. These are some of the main motivations for writing this book. Even though there are references to many mathematical areas in this book, it is practically self-contained, and is intended for the use by a wide audience of people interested in the subject.

Basic definitions and background
We will start with definitions and overview of the main mathematical objects that are involved in the inverse problems of our interest. These include the domains of definitions of the functions and operators, the boundary and spectral data and interpolation/extrapolation and restriction techniques.

The inverse problems
Rectangular directed layered grid Rectangular grids and gluing graphs Ordinary differential equations (ODEs)
 * === Solving polynomial equation  ===
 * === Pascal triangle  ===
 * === Monodromy operator  ===
 * === Three-term recurrence matrices and continued fractions  ===

Transformations of embedded graphs
The rules for replacing conductors in series or parallel connection by a single electrically equivalent conductor follow from the equivalence of the Y-Δ or star-mesh transforms.

Blaschke products
Let a_i be a set of n points in the complex unit disc. The corresponding Blaschke product is defined as


 * $$B(z)=\prod_k\frac{|a_k|}{a_k}(\frac{a_k-z}{1-\overline{a_k}z}).$$

If the set of points is finite, the function defines the n-to-1 map of the unit disc onto itself,


 * $$f:\mathbb{D}\xrightarrow[]{n\leftrightarrow 1}\mathbb{D}.$$

If the set of points is infinite, the product converges and defines an automorphism of the complex unit disc, given the Blaschke condition


 * $$\sum_k (1-|a_k|) <\infty.$$

The following fact will be useful in our calculations:


 * $$B(0)=\prod_n{|a_i|}.$$

Cauchy matrices

 * The Cayley transform provides the link between the Stieltjes continued fractions and Blaschke products and the Pick-Nevanlinna interpolation problem at the unit disc and the half-space.

Solution of the inverse problem
Rotation invariant layered networks

A. Elementary symmetric functions and permutations B. Continued fractions and interlacing properties of zeros of polynomials C. Wave-particle duality and identities involving integrals of paths in a graph and its Laplacian eigenvalues D. Square root and finite-differences

Given the Dirichlet-to-Neumann map of a layered network, find the eigenvalues and the interpolate, calculate the Blaschke product and continued fraction. That gives the conductivities of the layeres.

The square root of the minus Laplacian

 * We will now consider an important special case of the inverse problem

Notation


\mathbb{R} \mbox{ is the set of real numbers} $$



\mathbb{R}^N \mbox{ is the N-dimensional Euclidean space} $$



\mathbb{C} \mbox{ is the set of complex numbers} $$



\mathbb{D}=\{z \in \mathbb{C}|, |z| \le 1\} \mbox{ is the closed unit disc} $$



M \mbox{ is surface} $$



\alpha, \beta, \ldots \mbox{ are analytic functions} $$



\nabla \mbox{ is gradient} $$



\Delta \mbox{ Laplace operator} $$



\Lambda \mbox{ is Dirichlet-to-Neumann operator} $$



A, B, \ldots \mbox{ are matrices} $$



\lambda \mbox{ is eigenvalue} $$



\rho \mbox{ is characteristic polynomial} $$



P \mbox{ is permutation matrix} $$



F \mbox{ is Fourier transform} $$



\Gamma \mbox{ is graph} $$



G \mbox{ is network} $$



\gamma \mbox{ is conductivity} $$



q \mbox{ is potential} $$

Acknowledgements
The author would like to thank Wiki project for the help in all stages of writing the book.