  We consider a particle undergoing diffusion with stochastic resetting in a
bounded domain $\calU\subset \R^d$ for $d=2,3$. The domain is perforated by a
set of partially absorbing targets within which the particle may be absorbed at
a rate $\kappa$. Each target is assumed to be much smaller than $|\calU|$,
which allows us to use asymptotic and Green's function methods to solve the
diffusion equation in Laplace space. In particular, we construct an inner
solution within the interior and local exterior of each target, and match it
with an outer solution in the bulk of $\calU$. This yields an asymptotic
expansion of the Laplace transformed flux into each target in powers of
$\nu=-1/\ln \epsilon$ ($d=2$) and $\epsilon$ ($d=3$), respectively, where
$\epsilon$ is the non-dimensionalized target size. The fluxes determine how the
mean first-passage time to absorption depends on the reaction rate $\kappa$ and
the resetting rate $r$. For a range of parameter values, the MFPT is a unimodal
function of $r$, with a minimum at an optimal resetting rate $r_{\rm opt}$ that
depends on $\kappa$ and the target configuration.
