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.

## 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 where is a given constant , is a given integrable function, is standard Brownian motion.

According to the section ( Representation of solution for elliptic PDE using stochastic process ), the function is also a solution of the following problem.

Problem

(Example strong elliptic problem) Find s.t.

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 s.t.

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

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

Problem

(Example approximate problem) Find s.t.

Remark

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

The procedure is performed in the script OTSProject/python/wavelet2/poisson.py. We chose For such choice we verify the answer directly: We apply the operation . Thus where the constants , are chosen to satisfy Thus or We add and subtract the equation,

Using 45 basis functions we achieve the precision