  Analytical continuation is a central step in the simulation of
finite-temperature field theories in which numerically obtained Matsubara data
is continued to the real frequency axis for physical interpretation. Numerical
analytic continuation is considered to be an ill-posed problem where
uncertainties on the Matsubara axis are aplified exponentially. Here, we
present a systematic and controlled procedure that approximates any Matsubara
function by a minimal pole representation to within a predefined precision. We
then show systematic convergence to the exact spectral function on the real
axis as a function of our control parameter for a range of physically relevant
setups. Our methodology is robust to noise and paves the way towards reliable
analytic continuation in many-body theory and, by providing access to the
analytic structure of the functions, direct theoretical interpretation of
physical properties.
