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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Vydáno v:	Linear algebra and its applications Ročník 496; s. 1 - 35
Hlavní autoři:	Gilbert, Anna C., Hoskins, Jeremy G., Schotland, John C.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Inc 01.05.2016
Témata:	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
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Popis
Shrnutí:	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