I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 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.
 1 Two-sided area of integration, positive a.
 2 Two-sided area of integration, negative a.
 3 Indexing integration domains.
 4 Summary. Calculation of scalar product in N dimensions.
 5 Indexing integration domains II.
 6 Scalar product in N-dimensions. Test case 1.
 7 Scalar product in N-dimensions. Test case 2.
 8 Scalar product in N-dimensions. Test case 3.
 9 Implementation of scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Indexing integration domains II.

t remains to describe a recipe for construction of the maps , , .

The map indexes a matrix. We choose the "row-by-row" indexing. If then Therefore, if then and

With so defined we can calculate the arrays in parallel.

Therefore, to identify the mappings and it suffices to map domains of these mappings into the domain of . For each from domain of we perform on a separate thread the following calculation: where the last set is the set of numbers in domain of . The total size of calculated data is known apriori after the arrays have been calculated. Hence, after we calculate each on a separate thread, we do not need to merge data because the collection can happen directly into correct place.

To be precise, let and is obtained via the procedure of the section ( Calculation of partial sums in parallel ). We allocate an array For every (thread index) we calculate as described above. We allocate the result into as follows. For the result is placed into For the result is placed into For any the result is placed into The result is the indexing of the boundary subdomains and is enumeration of the boundary subdomains.

The case is almost identical. We replace with Indeed, we are interested in the area We arrive to the following procedure. Allocate and, for every , place result of into . Then is the indexing of the internal subdomains and is enumeration of the internal subdomains.