  We calculate numerically the sizes S of jumps (avalanches) between
successively pinned configurations of an elastic line (d=1) or interface (d=2),
pulled by a spring of (small) strength m^2 in a random-field landscape. We
obtain strong evidence that the size distribution, away from the small-scale
cutoff, takes the form P(S) = p(S/S_m) <S>/S_m^2, where S_m:=<S^2>/(2<S>),
proportional to m^(-d-zeta), is the scale of avalanches, and zeta the roughness
exponent at the depinning transition. Measurement of the scaling function f(s)
:= s^tau p(s) is compared with the predictions from a recent Functional RG
(FRG) calculation, both at mean-field and one-loop level. The avalanche-size
exponent tau is found in good agreement with the conjecture tau = 2-
2/(d+zeta), recently confirmed to one loop via the FRG. The function f(s)
exhibits a shoulder and a stretched exponential decay at large s, with ln f(s)
proportional to - s^delta, and delta approximately 7/6 in d=1. The function
f(s), universal ratios of moments, and the generating function <exp(lambda s)>
are found in excellent agreement with the one-loop FRG predictions. The
distribution of local avalanche sizes, i.e. of the jumps of a subspace of the
manifold of dimension d', is also computed and compared to our FRG predictions,
and to the conjecture tau' =2- 2/(d'+zeta).
