  The interpretation of helioseismic measurements, such as wave travel-time, is
based on the computation of kernels that give the sensitivity of the
measurements to localized changes in the solar interior. These are computed
using the ray or the Born approximation. The Born approximation is preferable
as it takes finite-wavelength effects into account, but can be computationally
expensive. We propose a fast algorithm to compute travel-time sensitivity
kernels under the assumption that the background solar medium is spherically
symmetric. Kernels are typically expressed as products of Green's functions
that depend upon depth, latitude and longitude. Here, we compute the spherical
harmonic decomposition of the kernels and show that the integrals in latitude
and longitude can be performed analytically. In particular, the integrals of
the product of three associated Legendre polynomials can be computed thanks to
the algorithm of Dong and Lemus (2002). The computations are fast and accurate
and only require the knowledge of the Green's function where the source is at
the pole. The computation time is reduced by two orders of magnitude compared
to other recent computational frameworks. This new method allows for flexible
and computationally efficient calculations of a large number of kernels,
required in addressing key helioseismic problems. For example, the computation
of all the kernels required for meridional flow inversion takes less than two
hours on 100 cores.
