  We consider the accuracy of slender body theory in approximating the force
exerted by a thin fiber on the surrounding viscous fluid when the fiber
velocity is prescribed. We term this the slender body inverse problem, as it is
known that slender body theory converges to a well-posed PDE solution when the
force is prescribed and the fiber velocity is unknown. From a PDE perspective,
the slender body inverse problem is simply the Dirichlet problem for the Stokes
equations, but from an approximation perspective, nonlocal slender body theory
exhibits high wavenumber instabilities which complicate analysis. Here we
consider two methods for regularizing the slender body approximation: spectral
truncation and the $\delta$-regularization of Tornberg and Shelley (2004). For
a straight, periodic fiber with constant radius $\epsilon>0$, we explicitly
calculate the spectrum of the operator mapping fiber velocity to force for both
the PDE and the approximations. We show that the spectrum of the original
slender body approximation agrees closely with the PDE solution at low
wavenumbers but differs at high frequencies, allowing us to define a truncated
approximation with a wavenumber cutoff $\sim1/\epsilon$. For both the truncated
and $\delta$-regularized approximations, we obtain similar convergence results
to the PDE solution as $\epsilon\to0$: a fiber velocity with $H^1$ regularity
gives $O(\epsilon)$ convergence, while a fiber velocity with at least $H^2$
regularity yields $O(\epsilon^2)$ convergence. Moreover, we determine the
dependence of the $\delta$-regularized error estimate on the regularization
parameter $\delta$.
