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 inversion of Black equation.

his section documents a negative result.

We would like to construct a preconditioner via exact inversion of the Black PDE.

In the section ( Asymptotic expansion for Black equation ) we calculated solution for Black equation. Let be the solution of the problem for integrable function then We would like to put the expression into the form for some kernel . We introduce the notation then We make the change then

We introduce the projections of on some basis and index set . We approximate then we apply the operation to the equation and obtain Thus where the are columns and are matrixes

Note that and It follows from or We intend to use (some approximation of) as a preconditioner. We need some way to evaluate the integrals perhaps with relaxed precision. The simplest road is to use crudification operator to simplify piecewise polynomial representations of the basis , make square of the number of pieces below 256 and then employ Cuda.

The functions are piecewise quadratic polynomials. Our next task is to calculate expressions for the integrals We calculate the -integral: We make the change We complete the square: Thus We make the change

We cannot proceed with explicit evaluation of the -integral from this point. Hence, we perform a linear approximation of the function on the interval of interest : Make the change then We complete the square: and do linear approximation

We put our results together.

Summary

(Integration of fundamental solution for Black equation) We calculate approximation of the integral according to the following procedure and is calculated as follows where

Due to lack of absolute precision and the number of required numerical operations this approach does not offer improvement over combined application of recipes of the sections ( Diagonal preconditioner ) and ( Reduction to well conditioned form ). The author did not do a numerical experiment or verification of the above summary.