User:DVD206/On Some Inverse Problems

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On 2D Inverse Problems

Dedicated to Nicole DeLaittre


Summary [edit | edit source]

The inverse problems, which this book is about are the mathematical problems of recovering the coefficients of functional and differential systems of equations from data about their solutions. These problems are opposite in some sense to the forward problems of evaluating functions. The inverse problems are well suited for computer simulations and many classical and current mathematical problems can be restated with ease as inverse problems on graphs or manifolds. Also the context of the inverse problems provides a unified point of view on the work of many great mathematicians.

These are some of the man motivations for writing this book.

The study of inverse problems takes its roots from medical imaging, such as CT scans, X-rays and MRIs and oil & gas production industry. It was motivated by needs of non-destructive and non-intrusive methods for study of hidden objects such as human organs or Earth's natural resources.

The tools of study and solutions of the inverse problems considered in this book allow one to "see inside" the objects using data about the electro-magnetic fields and sound waves observed at the boundary or outside the object.

Even though we reference 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[edit | edit source]

Graphs and manifolds [edit | edit source]

Harmonic functions[edit | edit source]

On random processes [edit | edit source]

The inverse problems [edit | edit source]

Applications to classical problems [edit | edit source]

Rectangular directed layered grid

Rectangular grids and gluing graphs

Ordinary differential equations (ODEs)

On the inverse problem of Calderon [edit | edit source]

"Can One Hear the Shape of a Drum?" [edit | edit source]

On inhomogeneous string of Krein [edit | edit source]

Special matrices and determinants [edit | edit source]

Embedded graphs and their transformations[edit | edit source]

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 transforms[edit | edit source]

Medial graphs [edit | edit source]

Dual graphs and harmonic conjugates [edit | edit source]

Determining genus of a graph[edit | edit source]

Hamilton paths in graphs [edit | edit source]

The new spectral theorem [edit | edit source]

The layered case [edit | edit source]

Fourier coordinates [edit | edit source]

Stieltjes continued fractions [edit | edit source]

A finite continued fraction is an expression of the form

Blaschke products [edit | edit source]

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

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

where n is the number of points.

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

The following fact will be useful in our calculations:

Pick-Nevanlinna interpolation [edit | edit source]

Cauchy matrices [edit | edit source]

The Cayley transform provides the link between the Pick-Nevanlinna interpolation problem at the unit disc and the half-space.

Solution of the layered inverse problem [edit | edit source]

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.

User:Daviddaved/The square root of the minus Laplacian [edit | edit source]

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

The case of the unit disc [edit | edit source]

Zolotorev problem [edit | edit source]

One more example [edit | edit source]

Total positivity [edit | edit source]

Compound matrices [edit | edit source]

Variation diminishing property [edit | edit source]

Spectral properties [edit | edit source]

Electrical networks[edit | edit source]

Dirichlet-to-Neumann operator [edit | edit source]

Effective conductivities[edit | edit source]

Connections between discrete and continuous models [edit | edit source]

Kernel of Dirichlet-to-Neumann map [edit | edit source]

Riemann mapping theorem[edit | edit source]

Hilbert transform [edit | edit source]

Acknowledgements[edit | edit source]

The author would like to than Wikipedia for ... Many thanks to the students of the REU summer school on inverse problems at the UW.

Bibliography [edit | edit source]