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.
 7 Solving one dimensional Black PDE.
 A. Example Black equation parameters.
 B. Reduction to system of linear algebraic equations for Black PDE.
 C. Adaptive time step for Black PDE.
 D. Localization.
 E. Reduction to system of linear algebraic equations for q=1.
 F. Preconditioner for Black equation in case q=1.
 G. Summary for Black equation in case q=1.
 H. Implementation of Black equation solution.
 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.

## Adaptive time step for Black PDE.

he recipes of this section were not verified because we are using a procedure that makes the issue of time step selection insignificant. We will discover that selection of time step should be made based on the task of the section ( Rebalancing wavelet basis ). However, it is not uncommon to encounter a practical situation when every optimization must be made. Hence, we leave this discussion in place.

According to the proposition ( Convergence of discontinuous Galerkin technique ), we need to keep the product around a constant value for all .

There are several problems with such task. If we substitute with then we run into problem because -th derivative of is zero. We may differentiate -times and arrive to then we substitute for and replace with However, is a sum of wavelets and the wavelets have a piecewise polynomial representation of second degree for numerical stability reasons. Thus, taking -norm of second derivative is not convenient. Finally, we do not know until we actually make the time step.

In the section ( Asymptotic expansion for Black equation ) we calculate asymptotic of the solution. We might try to apply the operation to that. However, calculation of asymptotic of sufficient order requires taking high order derivatives with respect to .

The following procedure might work. We make a first time step at arbitrary small length. The we calculate and use one step finite difference to calculate using values for consecutive . Thus we get small error from using finite differences and another error from being late by one time period. We compensate by taking smaller general level of .

Another possibility is finite difference approximation for after time steps.