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.
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Adapting wavelets to the interval [0,1].

dapting scaling functions to the interval [0,1] is based on the technique of the section ( Adapting dual wavelets to interval [0,1] ) and remarks of the previous section. The python code is located in the file "OTSProjects\python\wavelet\interval\interval.py".

The classes PsiGenerator and PsiGenerator2 implement the procedure. The script "_run_interval.py" performs the calculations.

We modify the procedure to reflect the fact that the functions MATH are calculated with significantly better precision than the functions MATH . We replace the formulas ( Wavelets on 01 step 1 ) with the following procedure that does not involve MATH .

Note that the formulas ( Wavelets on 01 step 1 ) are of the form MATH where the MATH and MATH are biorthogonal bases. The goal is to do without MATH . Let $v$ be the orthogonal projection of $u_{0}$ on the linear span of the finite set MATH . Thus MATH for some numbers MATH . We apply the operation MATH and obtain MATH In matrix notation, MATH If the set MATH is linearly independent then the matrix MATH is non-degenerate. In addition, MATH where MATH Thus MATH and we recover the MATH for the formula MATH by solving the system MATH

After we recover the boundary functions MATH we do not perform biorthogonalization because we cannot (the MATH have poor precision) but also because there is no need. Indeed, to conduct finite element calculations we need linear independence, approximation properties and subspace decomposition stability. But then, in light of the remark ( Dimension mismatch ), we need to separate correct number of basis functions. This is the motivation for the following sections.

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Copyright 2007