A fast algorithm for solution of a scattering problem using a recursive aggregate τ matrix method

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Abstract
An algorithm based on the recursive operator algorithm is proposed to solve for the scattered field from an arbitrarily shaped, inhomogeneous scatterer. In this method, the scattering problem is first converted to an N‐scatterer problem. Then, an add‐on procedure is developed to obtain recursively an (n + 1)‐scatterer solution from an n‐scatterer solution by introducing an aggregate τ matrix in the recursive scheme. The nth aggregate τ(n) matrix introduced is equivalent to a global τ matrix for n scatterers so that in the next recursion, only the two‐scatterer problem needs to be solved: One scatterer is the sum of the previous n scatterers, characterized by an nth aggregate τ(n) matrix; the other is the (n + 1)th isolated scatterer, characterized by τn + 1(1). If M is the number of harmonics used in the isolated scatterer T matrix and P is the number of harmonics used in the translation formulas, the computational effort at each recursion will be proportional to P2M. (Here we assume M is less than P.) Consequently, the total computational effort to obtain the N‐scatterer aggregate τ(N) matrix will be proportional to P2MN. In the low‐frequency limit, the algorithm is linear in N because P, the number of the harmonics in the translation formulas, is independent of the size of the object.

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