User:Daviddaved/On 2D Inverse Problems

From Wikibooks, open books for an open world
Jump to: navigation, search
On Inverse Problems in 2D

Dedicated to Nicole DeLaittre

About the book [edit]

Graph and dual.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 background[edit]

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.

Graphs and manifolds [edit]

Harmonic functions[edit]

On random processes [edit]

Special matrices and determinants [edit]

Electrical networks[edit]

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 Krein[edit]

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 transforms[edit]

Medial graphs [edit]

Dual graphs and harmonic conjugates[edit]

Determining genus of a graph[edit]

Hamilton paths in graphs[edit]

The new spectral theorem [edit]

The layered case [edit]

Fourier coordinates [edit]

Stieltjes continued fractions [edit]

Blaschke products[edit]

Let a_i be a set of n 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,

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:


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 at the unit disc and the half-space.

Solution of the inverse problem [edit]

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 [edit]

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

The case of the unit disc [edit]

Zolotarev problem [edit]

One more graph example[edit]

Connections between discrete and continuous models [edit]

Kernel of Dirichlet-to-Neumann map[edit]

Riemann mapping theorem[edit]

Hilbert transform [edit]

Schrodinger equation [edit]

Variation diminishing property [edit]

Spectral properties [edit]

Notation [edit]

\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 [edit]

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

About the author [edit]

Bibliography [edit]

  1. Astala, K. and P¨aiv¨arinta, L. "Calder´on’s inverse conductivity problem in the plane",
  2. Biesel, O. D., Ingerman D. V., Morrow J. A. and Shore W. T. "Layered Networks, the Discrete Laplacian, and a Continued Fraction Identity",
  3. Borcea, L. , Druskin, V. and Knizhnerman, L. "On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids", Communications on Pure and Applied Mathematics, Vol. 000, 0001–0048 (2000)
  4. Borcea, L., Druskin, V., Vasquez, G. F. and Mamonov, A. V. "Resistor network approaches to electrical impedance tomography",
  5. Cannon, J. W. "The combinatorial Riemann mapping theorem",
  6. Curtis, E. B. and Morrow, J. A. "Inverse Problems for Electrical Networks",
  7. Curtis, E. B., Ingerman, D. V. and Morrow, J. A. "Circular Planar Graphs and Resistor Networks",
  8. De Verdière, Yves Colin, "Reseaux electrique planaires I",
  9. Doyle, P. G. and Snell, L. J. "Random walks and electric networks",
  10. Edelman, A. and Strang, G. "Pascal matrices",
  11. Fomin, S. "Loop-erased walks and total positivity",
  12. Gantmacher, F. R. and Krein, M. G. "Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems", Revised Edition,
  13. Ingerman, D. V. "The Square of the Dirichlet-to-Neumann map equals minus Laplacian",
  14. Kac, M. "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23, (1966), doi:10.2307/2313748
  15. Karlin, S. "Total positivity",
  16. Kohn, R. and Vogelius, M. "Determining conductivity by boundary measurements", Comm. Pure Appl. Math., 37(1984), 289–298,
  17. Marshall, D. E. "An elementary proof of the Pick-Nevanlinna interpolation theorem",
  18. Nachman, A. I. "Global uniqueness for a two-dimensional inverse boundary value problem",
  19. Petrushev, P. P. and Popov, A. V. "Rational approximation of real functions", 1987,
  20. Pinkus, A. "Totally positive matrices",
  21. Sylvester, J. and Uhlmann, G. "A global uniqueness theorem for an inverse boundary value problem", Ann. of Math., 125 (1987), 153–169.
  22. Uhlmann, G. "Electrical impedance tomography and Calder´on’s problem",