Fast, Exact, and Stable Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation
抄録
We develop exact expressions for translations and rotations of local and
multipole fundamental solutions of the Helmholtz equation in spherical
coordinates. These expressions are based on recurrence relations that we
develop, and to our knowledge are presented here for the first time. The
symmetry and other properties of the coefficients are also examined, and
based on these efficient procedures for calculating them are presented. Our
expressions are direct, and do not use the Clebsch-Gordan coefficients or
the Wigner 3-j symbols, though we compare our results with methods that use
these, to prove their accuracy. We test our expressions on a number of
simple calculations, and show their accuracy. For evaluating a $N_t$ term
truncation of the translation (involving $O(N_t^2)$ multipoles), compared to
previous exact expressions that rely on the Clebsch-Gordan coefficients or
the Wigner $3-j$ symbol that require $O(N_t^5)$ operations, our expressions require $O(N_t^4)$) evaluations, with a small constant multiplying the order
term.
The recent trend in evaluating such translations has been to use approximate
"diagonalizations," that require $O(N_t^3)$ evaluations with a large
coefficient for the order term. For the Helmholtz equation, these
translations in addition have stabilty problems unless the accuracy of the
truncation and approximate translation are balanced. We derive explicit
exact expressions for achieving "diagonal" translations in $O(N_t^3)$
operations. Our expressions are based on recursive evaluations of multipole
coefficients for rotations, and are accurate and stable, and have a much
smaller coeffiicient for the order term, resulting practically in much fewer
operations. Future use of the developed methods in computational acoustic
scattering, electromagnetic scattering (radar and microwave), optics and
computational biology are expected.
Cross-referenced as UMIACS-TR-2001-44