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.
 8 Solving one dimensional mean reverting equation.
 A. Reduction to system of linear algebraic equations for mean reverting equation.
 B. Evaluating matrix R.
 C. Localization for mean reverting equation.
 D. Implementation for 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.

## Implementation for mean reverting equation. e calculate the matrix as described in the summary ( Summary for mean reverting equation in case q=1 ) and then apply manipulations of the summary ( Summary for Black equation in case q=1, inverted matrix ). We compare the results with the formula ( Analytical solution for mean reverting equation ). The difference is the fourth digit, consistently with the precision of the section ( Rebalancing wavelet basis section ) that we use when removing insignificant basis functions.

All scripts are located in the directory OTSProjects/python/wavelet2. The localization and gram matrix calculations are implemented in the scripts mrGram.py and mrMrGram.py. Initial grid calculation, see the section ( Decomposition of payoff function in one dimension ), is in the script mrGrid.py. The whole procedure is put together in the script mrFinal.py.

We calculate solution for the following parameters: Initial grid consists of 76 basis points but then calculations of the section ( Rebalancing wavelet basis section ) reveal that we need no more than 42.

The following figures ( Mean reversion 0 ) to ( Mean reversion 5 ) depict evolution of the solution as time changes from to . The slight instability on the figure ( Mean reversion 5 ) indicates that we do not have enough basis functions at the area where the solution changes fast. Comparison with the analytical solution ( Analytical solution for mean reverting equation ) indicates that the precision in the central area remains good. Plot of . Plot of . Plot of . Plot of . Plot of . Plot of calculated with 24 basis functions.