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

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

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 $\Delta t_{n}=1$ , the matrix $U$ has the following metrics: MATH The matrix $H$ has the following metrics: MATH If $\Delta t_{n}=0.1$ then the matrix $H$ has the following metrics: MATH If $\Delta t_{n}=100$ then the matrix $H$ has the following metrics: MATH Note that even for $\Delta t_{n}=100$ the spectral radius stays below $1$ . In addition, the Frobenius norm MATH decreases if $p$ increases and the maximal singular value drops below 1 for reasonably large $p$ .





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