It is possible to obtain the canonical basis from the PBW basis, as long as you are working in the quantum group U_q(n). The canonical basis seems to be inherently a quantum phenomenon, so it shouldn't be surprising that we want to compute in the quantum group.

If E_π is an indexing of the quantum PBW basis, then what one first proves is that applying the bar involution is unitriangular with respect to this basis. There is even a mathoverflow question about this: Convex PBW bases.

Now it is a matter of linear algebra to prove that there exists a unique bar invariant basis b_π such that $$b_\pi=\sum_\sigma c_{\pi\sigma} E_\sigma$$ with c_σσ=1, c_σπ∈qℤ[q] if σ≠π and c_σπ=0 unless σ≤π. This basis b_π is the canonical basis.

It is possible to obtain the canonical basis from the PBW basis, as long as you are working in the quantum group U_q(n). The canonical basis seems to be inherently a quantum phenomenon, so it shouldn't be surprising that we want to compute in the quantum group.

If E_π is an indexing of the quantum PBW basis, then what one first proves is that applying the bar involution is unitriangular with respect to this basis. There is even a mathoverflow question about this: Convex PBW bases.

Now it is a matter of linear algebra to prove that there exists a unique bar invariant basis b_π such that $$b_\pi=\sum_\sigma c_{\pi\sigma} E_\sigma$$ with c_σσ=1, c_σπ∈qℤ[q] if σ≠π and c_σπ=0 unless σ≤π. This basis b_π is the canonical basis.