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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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Solving one dimensional elliptic PDEs.


e apply technique of the section ( Finite elements ) and results of the section ( Calculation of approximation spaces in one dimension ) to the following problem.

Problem

(Example stochastic elliptic problem) Calculate MATH where $L$ is a given constant $L>0$ , MATH is a given integrable function, MATH $W_{t}$ is standard Brownian motion.

According to the section ( Representation of solution for elliptic PDE using stochastic process ), the function $u\left( x\right) $ is also a solution of the following problem.

Problem

(Example strong elliptic problem) Find MATH s.t. MATH

According to the section ( Elliptic PDE ), solution of the problem ( Example strong elliptic problem ) may be calculated by solving the weak problem.

Problem

(Example weak elliptic problem) Find MATH s.t. MATH

Following recipes of the section ( Finite elements for Poisson equation with Dirichlet boundary conditions ) we seek the solution of the form MATH where MATH is a selection from a wavelet basis on MATH . We take $v=w_{p}$ for $w_{p}$ from the same selection and arrive to the following approximate problem. MATH

Selection process for the basis MATH is already introduced in the section ( Decomposition of payoff function in one dimension ).

Problem

(Example approximate problem) Find MATH s.t. MATH

Remark

The scripts gramTest.py and gramTest2.py (placed in the directory OTSProjects/python/wavelet2) show that normalization in $W_{2}^{1}$ leads to a $W_{2}^{1}$ -Gram matrix with reasonable condition number for all scales $d$ and similar statement is true in $L_{2}$ . However, condition number of $W_{2}^{1}$ -Gram matrix of $L_{2}$ -normalized basis rapidly increases with $d$ -scale. The section ( Finite elements for Poisson equation with Dirichlet boundary conditions ) suggests that the $L_{2}$ -approximation of the input data $f$ is sufficient. Therefore, we select the basis MATH while approximating $f$ in $L_{2}[-1,1]$ -norm but then we renormalize the basis in MATH .

The procedure is performed in the script OTSProject/python/wavelet2/poisson.py. We chose MATH For such choice we verify the answer directly: MATH We apply the operation $\int_{0}^{x}dx$ . MATH MATH Thus MATH where the constants $C_{0}$ , $C_{1}$ are chosen to satisfy MATH Thus MATH or MATH We add and subtract the equation, MATH MATH MATH

Using 45 basis functions MATH we achieve the precision MATH





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