I have asked my question on Mathematics site of Stack Exchange but maybe I will get the answer rather here.
I am working on inverse problem - calibration of local volatility (financial application). This inverse problem is ill-posed. The optimization algorithm takes a $ \sigma $ surface as input and solves a PDE to calculate the cost function for each input. After inputs are evolved following genetic algorithm rules. There are several authors proposing next approaches to regularize the input surface:
- Regularization by means of Tikhonov
- Smoothing using B-splines
- Regularization using multiscale approach
Some authors use on of the first two approaches. However, the author of my article uses both of them(Main Article). For example the smoothness norm is introduced as:
$ \|\sigma\|^2 = \sum_{i}^{T} \int_{-A}^{A} dx \int_{0}^{T}|\frac{d\sigma}{dT}|^2 + |\frac{d^2\sigma}{dx^2}|^2 dx$
or in matrix form:
$ \|\sigma\|^2 = \theta A \theta^{T} $
where
$ \theta $ is a matrix of control points in B-Spline representation and
$ \sigma(T_{i},\cdot) = \sum_{m=1}^{M} \theta(i,m)\phi_{m}(\cdot) $
$ \sigma(t,x) = \frac{T_{i+1}-t}{T_{i+1}-T_{i}}\sigma(T_{i},x)+\frac{t-T_{i}}{T_{i+1}-T_{i}}\sigma(T_{i+1},x) $
are spline and linear interpolations by $x$ and $T$ axis respectively. After the authors use Cholesky decomposition to obtain matrix $B$ such that $ A = BB^{T} $. Finally they use genetic optimization algorithm to perform random changes of control points: $ \theta_{new} = \theta + B\epsilon_{i} $.
The PDE is solved again with new candidate solutions $ \theta_{new} $ until the stopping criteria is not achieved.
Basically I do not understand the difference between regularization and smoothing.
What I do understand:
B-Splines allow us to obtain a smooth representation of a function of interest
Regularization is putting some convenient constraint on the function to be optimized (including the constraint of smoothness).
Isn't it enough to introduce a spline representation of the surface? (won't it be already smooth - as I understand it will).
Why do not we apply the random change directly on $ \theta $? kind of $\theta_{new} = \theta+\epsilon_{i}$
The role of smoothness norm knowing that B-Spline function is apriori smooth is not clear for me. Thanks!