I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 1 Calculation of boundary scaling functions.
 2 Calculation of boundary wavelets.
 3 Testing properties of boundary wavelets and scaling functions.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 V. Stochastic optimization in one dimension.
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Calculation of approximation spaces in one dimension.

pplication of finite element technique to multidimensional problems suffers from curse of dimensionality. Our silver bullet for such curse is combined use of sparse tensor product (see the section ( Sparse tensor product )), high order wavelets (see the section ( Wavelet analysis )) and adaptive grid (see the section ( Adaptive approximation )).

The scaling functions , calculated in the section ( Calculation of scaling functions ) are approximations. Recall that we perform the procedure

starting from some and stop after a finite number of steps . We use , as approximations for . Hence instead of the desired relationship ( Scaling equation ): Therefore, we would like to project on linear span of . The size of the difference is indication of success of the procedure.

So far we have calculated wavelets on the interval (see the sections ( Adapting scaling function to the interval [0,1] ) and ( Adapting wavelets to the interval [0,1] )). If we attempt to use the functions , as a basis of sparse tensor product then we must be able to evaluate projection for all and the difference between the projection and the original in norm should be small. For calculation of the projection to be affective, the Gram matrix must have a low condition number for all . The script tCondNumber.py shows that the Gram matrix has condition number below 6 for any . Therefore, our goal is to pick boundary functions that do not significantly increase the condition number of the whole matrix. The boundary wavelets are chosen with the same consideration.

 1 Calculation of boundary scaling functions.
 2 Calculation of boundary wavelets.
 3 Testing properties of boundary wavelets and scaling functions.