Diffuse scattering on graphs

We formulate and analyze difference equations on graphs analogous to time-independent diffusion equations arising in the study of diffuse scattering in continuous media. Moreover, we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous (background p...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 496; pp. 1 - 35
Main Authors: Gilbert, Anna C., Hoskins, Jeremy G., Schotland, John C.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2016
Subjects:
Discrete mathematics in relation to computer science
Equations of mathematical physics and other areas of application
Graph algorithms
Graphs and groups
Graphs and matrices
68R
Equations of mathematical physics and other areas of application
05C50
05C85
Graphs and groups
Graphs and matrices
Graph algorithms
05C25
35Q
Discrete mathematics in relation to computer science
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:We formulate and analyze difference equations on graphs analogous to time-independent diffusion equations arising in the study of diffuse scattering in continuous media. Moreover, we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous (background problem) using Born series, providing necessary conditions for convergence and demonstrating the process through numerous examples. In addition, we outline a method for finding Green's functions for Cayley graphs for both abelian and non-abelian groups. Finally, we conclude with a discussion of the effects of sparsity on our method and results, outlining the simplifications that can be made provided that the scatterers are weak and well-separated.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.01.012