Consider a finite dimensional approximation to a function u :
.
We want to find an approximation to the first derivative w.r.t. x:
.
It is convenient to describe the principal idea for the two-dimensional case. Then, the multiindices read
and
. The index set is decomposed into 'lines' along the x-coordinate direction

i.e.

Then

The derivative of the term in the brackets [...] can be discretized using finite differences. The main steps are

- inverse wavelet transform to obtain the nodal values on an univariate grid with points depending on
- application of the univarite finite difference scheme
- wavelet transform with respect to the particular coordinate direction

The figure below is a graphical sketch of the scheme for the 2D case. The squares on the left hand side represent the wavelet space with a certain arrangement of the indices/wavelet coefficients. The coloured entries are the coefficients for the indices from (the colour corresponds to their magnitude). The white region corresponds to indices which are not in . In each step just one line (marked by the red bars; we have shown two different lines in this example) of coefficients is read and inverse transformed. This yields the nodal values of an univariate partial function on a (non-uniform) grid. The grid is dependent on the particular line. To these values the finite difference scheme is applied. Then, a wavelet transform yields the coefficients of the result. This repeats for all lines. Vertical lines would be read/written for derivatives with respect to the y-coordinate direction. An analysis of the resulting consistency error is given in [3] for regular sparse grids and in [5] for general adaptive grids.

In the same fashion, a multivariate inverse Interpolet transform is performed. First by an inverse transform along and then along . Note, this simple scheme for the transform works for Interpolets only, and not for e.g. Lifting Interpolets or the Daubechies wavelets! To explain why is a longer story ;-) !