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.

## Reduction to well conditioned form.

e noted in the section ( Diagonal preconditioner ) that the properties of the matrix of the system improve if we replace with powers of . The calculations of the section ( Analytical preconditioner derived from asymptotic decomposition in time ) hint on how we would use such observation.

We multiply the equation with : and obtain We multiply with : and continue

Summary

(Improving condition number) An equation of the form may be replaced with either or

The matrices may be evaluated via the procedure

Numerical experimentation in the script blackDp.py show that the condition number for matrix computed from improves as follows The maximal singular values and Frobenius norms of are This procedure works as a substitution for inversion.