I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 1 Decomposition of payoff function in one dimension. Adaptive multiscaled approximation.
 2 Constructing wavelet basis with Dirichlet boundary conditions.
 3 Accelerated calculation of Gram matrix.
 4 Adapting wavelet basis to arbitrary interval.
 5 Solving one dimensional elliptic PDEs.
 6 Discontinuous Galerkin technique II.
 A. Rebalancing wavelet basis.
 B. Reduction to system of linear algebraic equations.
 7 Solving one dimensional Black PDE.
 8 Solving one dimensional mean reverting equation.
 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.

## Rebalancing wavelet basis.

roblem

(Rebalancing wavelet basis) Let be a Hilbert space, is a dense set in and is a basis in : with the following properties:

1.

2. For any and any ,

A compact operator is given by its description on the set .

We start from a function , precision and an approximation where is a small index subset of and Proj is projection in on set .

We would like to find an efficient procedure for calculation of the set or some approximation of from above. Evaluation of is prohibitively expensive. The matrix is known.

Think of being an approximation to a hut function (put straddle). The is calculated in the section ( Decomposition of payoff function in one dimension ). We place a lot of high level wavelets around the singularity at the strike. When we evolve with Backward Kolmogorov's equation, the singularity disappears. Therefore, the weights placed on high order wavelets around the strike should redistribute to lower order wavelets.

We produce by taking with maximal weights . Those , not included in , would have a small weight if included in the sum. When we apply the transformation , the weights redistribute. The function gives us partial information about such redistribution.

We form the set

We form the set recursively via distinct unions The optimal function have to depend on diffusive properties of operator . However, due to iterative nature of this procedure, would work too. In addition, we add propagation into higher scales as follows:

We denote the construction of from as

We calculate Then we calculate We continue for We stop when becomes small.

After completion, the final set is reduced by removing indexes with lowest contribution. The result is .

Another possibility is simply to take to be a large number and do one iteration.