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.

## Reduction to system of linear algebraic equations for Black PDE.

e apply recipes of the section ( Reduction to system of linear algebraic equations ) to the problem ( Example strong Black problem 2 ).

We introduce the time mesh

The location of the points is to be chosen according to the statement ( Convergence of discontinuous Galerkin technique ) and will be addressed in the section ( Adaptive time step for Black PDE ).

First, we apply results of the section ( Decomposition of payoff function in one dimension section ) to the function of the problem ( Example strong Black problem 2 ). Thus, we have a wavelet basis and we start from decomposition for some index set

According to the results around the formula ( Matrix of discontinuous Galerkin ), we perform a single time step transformation by solving the equation where we introduce double indexes , , is a column , is a matrix For the present problem, the operator does not have -dependency. Hence and where the functions are second degree piecewise polynomials constructed to be in . The column comes from previous step or input data: Thus, for , we put and for the is taken from the previous step.

We introduce the notations Then We arrange indexes by changing the spacial indexes before changing the time indexes . Then the matrix has block structure. For we get

For any matrixes and , we introduce the notation " ": Note that

We get where the last line is valid for all .

For we calculate Let then We obtain Note that is a strongly diagonal matrix and is a well conditioned matrix. We act on the last equation by and obtain

Summary

(Reduction of Black equation) We introduce the time mesh and seek a solution of the problem ( Example strong Black problem 2 ) in the form where is a wavelet basis for constructed in the section ( Adapting wavelet basis to arbitrary interval ).

Then and is derived from recursively for via We use the notations

The index sets come from considerations of the sections ( Rebalancing wavelet basis ) and ( Decomposition of payoff function in one dimension ). The time steps come from the section ( Adaptive time step for Black PDE ).

The summary above shows that if the operator is -independent then we can solve the problem with using the same manipulations as in the case without noticeable increase in memory requirements. The space and time parts of the problem may be processed separately. This observation extends to a situation when there is multiplicative separation of space and time dependent expressions in the PDE, for example