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.
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## Diagonal preconditioner.

e multiply the equation of the summary ( Reduction to system of linear algebraic equations for q=1 ) with a diagonal matrix : and transform the matrix or We would like to choose the matrix so that the Frobenius norm of the matrix would be minimal.

We calculate thus We observe that is a positive quadratic function of , thus the minimum is located where the derivatives vanish. We differentiate and transform

Calculation of the script blackDp.py in the directory OTSProject/python/wavelet2 shows that such preconditioner is very effective. For the parameters, set in the section ( Example Black equation parameters ) and , the matrix has the following metrics: The matrix has the following metrics: If then the matrix has the following metrics: If then the matrix has the following metrics: Note that even for the spectral radius stays below . In addition, the Frobenius norm decreases if increases and the maximal singular value drops below 1 for reasonably large .

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 Copyright 2007