What would we like to obtain?
On the one hand, the model prices for the forward local volatility problem (e.g. by finite elements) should fit the (noisy) market prices reasonably, on the other hand, the local volatility surface should not exhibit severe oscilations. It should be as smooth as possible and as unsmooth as necessary to fit the data.
Identifying local volatility is, at least in infinite dimensional setting (you would know the call price for any (K, T)-combination (K is strike, T is expiry)) an ill-posed problem. This means that arbitrarily small noise (remark; if of sufficient high frequency) can lead to arbitrarily large errors in the solution. And it is a nonliner problem, which makes it more complicated than, e.g. curvefitting in a Hull-White-setting.
The technique to overcome these stability prblems (or: one of the techniques) is Tikhonov regularisation. The following plot shows the result we obtain by regularization on the same data as we used in the presmoothing example.
