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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.
A. Example Black equation parameters.
B. Reduction to system of linear algebraic equations for Black PDE.
C. Adaptive time step for Black PDE.
D. Localization.
E. Reduction to system of linear algebraic equations for q=1.
F. Preconditioner for Black equation in case q=1.
G. Summary for Black equation in case q=1.
H. Implementation of Black equation solution.
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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Reduction to system of linear algebraic equations for Black PDE.


e apply recipes of the section ( Reduction to system of linear algebraic equations ) to the problem ( Example strong Black problem 2 ).

We introduce the time mesh MATH

The location of the points $\tau$ is to be chosen according to the statement ( Convergence of discontinuous Galerkin technique ) and will be addressed in the section ( Adaptive time step for Black PDE ).

First, we apply results of the section ( Decomposition of payoff function in one dimension section ) to the function $g\left( x\right) $ of the problem ( Example strong Black problem 2 ). Thus, we have a wavelet basis MATH and we start from decomposition MATH for some index set $K_{N}\subset P.$

According to the results around the formula ( Matrix of discontinuous Galerkin ), we perform a single time step transformation by solving the equation MATH where we introduce double indexes MATH , MATH , $c$ is a column MATH , MATH is a matrix MATH MATH For the present problem, the operator $B\left( t\right) $ does not have $t$ -dependency. Hence MATH and MATH where the functions MATH are second degree piecewise polynomials constructed to be in MATH . The column MATH comes from previous step or input data: MATH Thus, for $n=N-1$ , we put $v_{n+1}:=v_{N}$ and for $n<N-1$ the $v_{n+1}$ is taken from the previous step.

We introduce the notations MATH Then MATH We arrange indexes MATH by changing the spacial indexes $s,k$ before changing the time indexes $r,s$ . Then the matrix MATH has block structure. For $q=2$ we get MATH

For any matrixes $v$ and $G$ , we introduce the notation " $\times$ ": MATH Note that MATH

We get MATH where the last line is valid for all $q$ .

For MATH we calculate MATH Let MATH then MATH We obtain MATH Note that MATH is a strongly diagonal matrix and MATH is a well conditioned matrix. We act on the last equation by MATH and obtain MATH

Summary

(Reduction of Black equation) We introduce the time mesh MATH and seek a solution $v$ of the problem ( Example strong Black problem 2 ) in the form MATH where MATH is a wavelet basis for MATH constructed in the section ( Adapting wavelet basis to arbitrary interval ).

Then MATH and MATH is derived from MATH recursively for $n=0,1,...,N-1$ via MATH MATH MATH We use the notations MATH MATH

The index sets $K_{n}$ come from considerations of the sections ( Rebalancing wavelet basis ) and ( Decomposition of payoff function in one dimension ). The time steps $\Delta t_{n}$ come from the section ( Adaptive time step for Black PDE ).

The summary above shows that if the operator $B\left( t\right) $ is $t$ -independent then we can solve the problem with $q>1$ using the same manipulations as in the case $q=1$ without noticeable increase in memory requirements. The space and time parts of the problem may be processed separately. This observation extends to a situation when there is multiplicative separation of space and time dependent expressions in the PDE, for example MATH





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