User:Daviddaved/On Inverse Problems in 2D

On Inverse Problems in 2D


Dedicated to Nicole DeLaittre


About the book Edit

Book cover.jpg

Summary Edit

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 backgroundEdit

This chapter gives the definitions and the 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.

Graphs and manifolds Edit

Harmonic functionsEdit

On random processes Edit

Special matrices and operators Edit

Electrical networksEdit

The inverse problems Edit

Rectangular directed layered grid

Rectangular grids and gluing graphs

Ordinary differential equations (ODEs)

Applications to classical problems Edit

On the inverse problem of Calderon Edit

"Can One Hear the Shape of a Drum?" Edit

On inhomogeneous string of KreinEdit

Transformations of embedded graphs Edit

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.

Y-Δ and star-mesh transformsEdit

Medial graphs Edit

Dual graphs and harmonic conjugatesEdit

Determining genus of a graphEdit

Hamiltonian paths in graphsEdit

The new spectral theorem Edit

Rotation invariant layered case Edit

Fourier coordinates Edit

Stieltjes continued fractions Edit

Blaschke productsEdit

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|}.

Pick-Nevanlinna interpolation Edit

Cauchy matrices Edit

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

Solution of the inverse problem Edit

A. Elementary symmetric functions and permutations B. Continued fractions and interlacing properties of zeros of polynomials C. Wave-particle duality and identities involving path integrals and Laplacian eigenvalues of a graph 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 Edit

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

The case of the unit disc Edit

One more exampleEdit

Zolotarev problem Edit

Discrete and continuous models Edit

Kernel of Dirichlet-to-Neumann mapEdit

Riemann mapping theoremEdit

Hilbert transform Edit

Schrodinger equation Edit

Variation diminishing property Edit

Spectral properties Edit

Notation Edit

Last modified on 20 October 2012, at 23:58