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.

## 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 where Thus, we transform into the set of coefficients such that is an approximation to the solution of the problem taken at :

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

It is essential that are constructed as piecewise quadratic functions in , 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 consider the transformation where the might be outside of span . The sum should be understood as a projection. We put together the matrix Hence, the semi-solution would act Thus The is represented by where is the -Gram matrix and is a column .

Thus we arrive to the chain Therefore, we set

Summary

(Analytic preconditioner for Black equation) The preconditioner for the equation is given by

The matrix is calculated via the following procedure. Let be the transformation For every we calculate the coefficients and form the matrix

The projection in the above summary is calculated as follows.

We introduce the convenience notation : Then the coefficients come from the relationships for the scalar product . Thus or The expression for the preconditioner is thus

It remains to improve the calculation of : where the functions are piecewise polynomials in vanishing at the ends of integration interval. We integrate by parts: and arrive to the same expression we already use when constructing the matrix . Thus while the matrix is given by Hence the preconditioned matrix is The matrix is symmetric.