  We develop an algorithm for the asymptotically fast evaluation of layer
potentials close to and on the source geometry, combining Geometric Global
Accelerated QBX (`GIGAQBX') and target-specific expansions. GIGAQBX is a fast
high-order scheme for evaluation of layer potentials based on Quadrature by
Expansion (`QBX') using local expansions formed via the Fast Multipole Method
(FMM). Target-specific expansions serve to lower the cost of the formation and
evaluation of QBX local expansions, reducing the associated computational
effort from $O((p+1)^{2})$ to $O(p+1)$ in three dimensions, without any
accuracy loss compared with conventional expansions, but with the loss of
source/target separation in the expansion coefficients. GIGAQBX is a `global'
QBX scheme, meaning that the potential is mediated entirely through expansions
for points close to or on the boundary. In our scheme, this single global
expansion is decomposed into two parts that are evaluated separately: one part
incorporating near-field contributions using target-specific expansions, and
one part using conventional spherical harmonic expansions of far-field
contributions, noting that convergence guarantees only exist for the sum of the
two sub-expansions. By contrast, target-specific expansions were originally
introduced as an acceleration mechanism for `local' QBX schemes, in which the
far-field does not contribute to the QBX expansion. Compared with the
unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated
cost model that the combined scheme yields a considerable cost reduction for
the near-field evaluation part of the computation. We support the effectiveness
of our scheme through numerical results demonstrating performance improvements
for Laplace and Helmholtz kernels.
