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.
 a. Analytical preconditioner derived from asymptotic decomposition in time.
 b. Diagonal preconditioner.
 c. Symmetrization and symmetric preconditioning.
 d. Reduction to well conditioned form.
 e. Analytical preconditioner derived from inversion of Black equation.
 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.

## Symmetrization and symmetric preconditioning.

n this section we document a negative result. We make a natural attempt to switch to a symmetrical problem and find that such transformation does not bring noticeable benefit.

We start from the equation of the section ( Diagonal preconditioner ) and multiply it with the operator . We obtain a system with symmetrical matrix: We aim for the matrixes and to be as small as possible. The symmetrization step took us in opposite direction.

Matrix for .

We note that we cannot apply the diagonal preconditioner of the section ( Diagonal preconditioner ) because such operation looses the symmetry. Hence, we attempt the following transformation: for some new diagonal matrix that we select from minimization of Frobenius norm. We calculate and we select by minimizing . Observe that then We seek the local minimum: We arrived to a system of linear equations: The symmetry is the only good feature of this system.

Matrix for .

The numerical experiment in the script blackDp.py shows that the condition number of for is 1883. The good thing is we do not need to be exact. However, given results of the non-symmetric line of calculation, this is not worth pursuing further.