Let $u_h$ be the $\mathbb{P}^k$ interpolant of some function $u$ (supposed smooth to simplify). By definition, $u_h$ writes $u_h=\sum_i u_i \varphi_i$, where $u_i$ is the value of $u$ at the point $i$ and $\varphi_i$ is the Lagrange polynomial associated to the point $i$.
Now, if I suppose that I choose to use not $u_i$ as defined above, but $\tilde{u}_i$ some (random maybe) value such that $||u_i-\tilde{u}_i||=o(h^k)$ and so I construct the approximation $\tilde{u}_h=\sum_i \tilde{u}_i \varphi_i$. The approximation $\tilde{u}_h$ is of the same order, but not interpolant.
Are there some situations where a non interpolant approximation like above (still using Lagrange basis functions) is preferred to the Lagrange interpolant ?