By Hassi S., Sebestyen Z., Snoo H.

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**Extra resources for A canonical decomposition for linear operators and linear relations**

**Sample text**

The inverse problem is to find the locations yl,y2, . . , y ~of the point scatterers from the knowledge of A . Directly follows the decomposition: A-H*TH (4) where H,l = ~ G ( 8 ~ , y mH )E, C M x N, H* E C N x Mis the adjoint matrix of H and T = diag ( s i g n 7,) E C M x M , s i g n T, := b m l . We define cpz as We mention that the direct use of data corresponding to point sources and for infinite domains is performed in Refs. 8,11. For our algorithm, the modification in this case corresponds to A = HT T H .

Is the main idea of our algorithm. , y ~ } This IT (b) 10" 6 1om 4 2 1oo 10 10 -5 Example 10 Reconstructed 5 point scatterers ( a ) contour plot, 0 (bj -10 3 - 0 plot 0 0 -10 5 Example 2 Reconstructed point scatterers, ( c ) contour plot, ( d ) 3-0plot In our examples the waveguide is a 2-dimensional horizontal strip +m) x (0, h) where h is the height of the waveguide, here h = 10. The wavelength is taken as X = 1 then k = 27r. In all the sums of Green's functions we have used approximately 2h 4 terms since in this case all propagating modes are taken into account and some evanescent.

For waveguides there are many results concerning inverse scattering techniques and applications to detect point scatterers. In Ref. 1 is given an application of a MUSIC type algorithm for a perturbed open waveguide for the detection of inclusions of small diameter. In Ref. 12, for two-dimensional acoustic waveguides with Dirichlet boundaries is given a mathematical justification using as incident fields eigenmodes. A far-field description is used. In this work, we formulate a MUSIC type algorithm for a simple waveguide and justify mathematically its connection with the scattering from an inhomogeneous waveguide.