  We consider two large polaron systems that are described by a Fr\"{o}hlich
type of Hamiltonian, namely the Bose-Einstein condensate (BEC) polaron in the
continuum and the acoustic polaron in a solid. We present ground-state energies
of these two systems calculated with the Diagrammatic Monte Carlo (DiagMC)
method and with a Feynman all-coupling approach. The DiagMC method evaluates up
to very high order a diagrammatic series for the polaron Green's function. The
Feynman all-coupling approach is a variational method that has been used for a
wide range of polaronic problems. For the acoustic and BEC polaron both methods
provide remarkably similar non-renormalized ground-state energies that are
obtained after introducing a finite momentum cutoff. For the renormalized
ground-state energies of the BEC polaron, there are relatively large
discrepancies between the DiagMC and the Feynman predictions. These differences
can be attributed to the renormalization procedure for the contact interaction.
