Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities

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.
a. Analytical preconditioner derived from asymptotic decomposition in time.
b. Diagonal preconditioner.
c. Symmetrization and symmetric preconditioning.
d. Reduction to well conditioned form.
e. Analytical preconditioner derived from inversion of Black equation.
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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Analytical preconditioner derived from asymptotic decomposition in time.

his section is of academic significance. To derive a preconditioner from asymptotic decomposition is a very natural attempt. The author feels compelled to document it even so the final result could be guessed. The reader may skip this section without consequence.

According to the section ( Reduction to system of linear algebraic equations for q=1 ), we need to solve the equation MATH where MATH Thus, we transform $v_{n+1}$ into the set of coefficients MATH such that MATH is an approximation to the solution MATH of the problem MATH taken at $t_{n}$ : MATH

In the section ( Asymptotic expansion for Black equation ) we calculated an approximate solution to the problem MATH in the form MATH Thus we have an explicit transformation MATH MATH where MATH is close to MATH . We use such transformation to construct a preconditioner MATH for the step MATH of the chain MATH .

It is essential that MATH are constructed as piecewise quadratic functions in MATH , see the sections ( Calculation of boundary scaling functions ), ( Calculation of approximation spaces in one dimension II ). Thus, the first derivative is continuous and piecewise linear function and the second derivative is a piecewise constant function. We do not reach a situation when we would have to keep track of delta functions. We can apply facilities of PiecewisePoly library (see the section ( Manipulation of localized piecewise polynomial functions )) without additional modification. It also helps to do integration by part to even out differentiation.

For every $k\in K_{n}~$ consider the transformation MATH where the MATH might be outside of span MATH . The sum should be understood as a projection. We put together the matrix MATH Hence, the semi-solution MATH would act MATH Thus MATH The $v_{n+1}$ is represented by MATH MATH where $G$ is the MATH -Gram matrix MATH and $x$ is a column MATH .

Thus we arrive to the chain MATH Therefore, we set MATH


(Analytic preconditioner for Black equation) The preconditioner $P^{(n)}$ for the equation $\left( \&\right) $ is given by MATH

The matrix $C$ is calculated via the following procedure. Let MATH be the transformation MATH For every $k\in K_{n}~$ we calculate the coefficients MATH MATH and form the matrix MATH

The projection in the above summary is calculated as follows.

We introduce the convenience notation $v_{k}$ : MATH Then the coefficients MATH come from the relationships MATH for the $L_{2}$ scalar product MATH . Thus MATH or MATH The expression for the preconditioner is MATH thus MATH

It remains to improve the calculation of $V$ : MATH where the functions $w_{k}$ are piecewise polynomials in $C^{1}$ vanishing at the ends of integration interval. We integrate by parts: MATH and arrive to the same expression we already use when constructing the matrix MATH . Thus MATH MATH while the matrix MATH is given by MATH Hence the preconditioned matrix is MATH The matrix $G$ is symmetric. MATH

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