User:DVD206/On 2D Inverse Problems
- 1 About the book
- 2 Summary
- 3 Basic definitions and background
- 4 Applications to classical problems
- 5 Transformations of embedded graphs
- 6 The layered case
- 7 The square root of the minus Laplacian
- 8 Connections between discrete and continuous models
- 9 Notation
- 10 Acknowledgements
- 11 About the author
- 12 Bibliography
- 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.
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.
- === Solving polynomial equation ===
Rectangular directed layered grid
- === Pascal triangle ===
Rectangular grids and gluing graphs
- === Monodromy operator ===
Ordinary differential equations (ODEs)
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.
- Rectangular grid.jpg
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,
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:
- 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.
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.
- We will now consider an important special case of the inverse problem
The author would like to thank Wiki project for the help in all stages of writing the book.
- Astala, K. and P¨aiv¨arinta, L. "Calder´on’s inverse conductivity problem in the plane", http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n1-p05.pdf
- Biesel, O. D., Ingerman D. V., Morrow J. A. and Shore W. T. "Layered Networks, the Discrete Laplacian, and a Continued Fraction Identity", http://www.math.washington.edu/~reu/papers/2008/william/layered.pdf
- 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)
- Borcea, L., Druskin, V., Vasquez, G. F. and Mamonov, A. V. "Resistor network approaches to electrical impedance tomography", http://arxiv.org/abs/1107.0343
- Cannon, J. W. "The combinatorial Riemann mapping theorem", http://www.springerlink.com/content/9w0608p039151254/
- Curtis, E. B. and Morrow, J. A. "Inverse Problems for Electrical Networks", http://books.google.com/books/about/Inverse_Problems_for_Electrical_Networks.html?id=gLnohh95zFIC
- Curtis, E. B., Ingerman, D. V. and Morrow, J. A. "Circular Planar Graphs and Resistor Networks", http://www.math.washington.edu/~morrow/papers/cim.pdf
- De Verdière, Yves Colin, "Reseaux electrique planaires I", http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.224.133
- Doyle, P. G. and Snell, L. J. "Random walks and electric networks", http://www.cse.buffalo.edu/~hungngo/classes/2005/Expanders/papers/general/randomWalk.pdf
- Edelman, A. and Strang, G. "Pascal matrices", http://web.mit.edu/18.06/www/pascal-work.pdf
- Fomin, S. "Loop-erased walks and total positivity", http://arxiv.org/pdf/math.CO/0004083.pdf
- Gantmacher, F. R. and Krein, M. G. "Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems", Revised Edition, http://www.ams.org/bookstore?fn=20&arg1=diffequ&ikey=CHEL-345-H
- Ingerman, D. V. "The Square of the Dirichlet-to-Neumann map equals minus Laplacian", http://arxiv.org/ftp/arxiv/papers/0806/0806.0653.pdf
- Kac, M. "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23, (1966), doi:10.2307/2313748
- Karlin, S. "Total positivity", http://books.google.com/books/about/Total_Positivity.html?id=yIarAAAAIAAJ
- Kohn, R. and Vogelius, M. "Determining conductivity by boundary measurements", Comm. Pure Appl. Math., 37(1984), 289–298, http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160370302/abstract
- Marshall, D. E. "An elementary proof of the Pick-Nevanlinna interpolation theorem", http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.mmj/1029001307&page=record
- Nachman, A. I. "Global uniqueness for a two-dimensional inverse boundary value problem", http://www.jstor.org/discover/10.2307/2118653?uid=3739960&uid=2&uid=4&uid=3739256&sid=21101022816693
- Petrushev, P. P. and Popov, A. V. "Rational approximation of real functions", 1987, http://books.google.com/books/about/Rational_Approximation_of_Real_Functions.html?id=0up9c_uo2xQC
- Pinkus, A. "Totally positive matrices", http://www.amazon.com/Totally-Positive-Matrices-Cambridge-Mathematics/dp/0521194083
- Sylvester, J. and Uhlmann, G. "A global uniqueness theorem for an inverse boundary value problem", Ann. of Math., 125 (1987), 153–169.
- Uhlmann, G. "Electrical impedance tomography and Calder´on’s problem", http://www.math.washington.edu/~gunther/publications/Papers/calderoniprevised.pdf