We compute roughness exponents of elastic d-dimensional manifolds in (d+1)-dimensional embedding spaces at the depinning transition for d=1, em leader,4. Our numerical method is rigorously based on a Hamiltonian formulation; it allows us to determine the critical manifold in finite samples for an arbitrary convex elastic energy. For a harmonic elastic energy (Delta(2) model), we find values of the roughness exponent between the one-loop and two-loop functional renormalization group results, in good agreement with earlier cellular automaton simulations. We find that the Delta(2) model is unstable with respect both to slight stiffening and to weakening of the elastic potential. Anharmonic corrections to the elastic energy allow us to obtain the critical exponents of the quenched Kardar, Parisi, Zhang class.