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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.
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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Accelerated calculation of Gram matrix.


e will be repeatedly calculating matrixes of the form MATH for some fixed functions $u$ and $v$ and a scalar product MATH in MATH . We use properties of scale and transport operations to reduce the number scalar product evaluations.

Let $d_{2}-d_{1}=d$ . We calculate MATH and use the formula ( Property of scale and transport 2 ). MATH We use the formula ( Property of scale and transport 3 ). MATH We use the formula ( Property of scale and transport 7 ). MATH Therefore, we regard the scalar product MATH as a function of two parameters $d_{2}-d_{1}$ and MATH . Together with the rule MATH this significantly reduces the amount of calculation.

In case of a wavelet basis, adapted to an interval $\left[ A,B\right] $ , we can apply this rule to the internal functions. MATH





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