Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities

I. Wavelet calculations.
1. Calculation of scaling filters.
2. Calculation of scaling functions.
3. Calculation of wavelets.
4. Convergence of cascade procedure.
5. Direct verification of wavelet properties.
6. Adapting scaling function to the interval [0,1].
7. Adapting wavelets to the interval [0,1].
II. Calculation of approximation spaces in one dimension.
III. Calculation of approximation spaces in one dimension II.
IV. One dimensional problems.
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.
Downloads. Index. Contents.

Convergence of cascade procedure.

he script in the file "_run_cascade.py" explores convergence of the scaling procedure for a variety of $\eta_{0}$ . We are trying to achieve acceptable precision of the final result $\phi$ or $\tilde{\phi}$ for one of the functions for a minimal number of steps of the scaling procedure. Minimization of the number of steps is important because every steps doubles the number of intervals in the piecewise polynomial description of the result.

For $n=5$ (the number of vanishing moments), numerical experimentation shows that MATH and MATH are optimal choices. The lower splines suffer from poor approximation. The higher splines struggle with numerical stability. The $L_{2}$ -difference between successive steps of the scaling procedure halves at every step. One of the scaling functions has the precision $0.001$ after 5 steps in case of MATH and precisions $0.003,0.0008$ after 4 and 5 steps in case of MATH . The choice MATH carries additional advantage of continuity of first derivative. For this reason we concentrate on such choice.

Downloads. Index. Contents.

Copyright 2007