I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 1 Crudification of piecewise-quadratic representation.
 2 Convergence of modified cascade procedure.
 3 Calculation of boundary scaling functions II.
 IV. One dimensional problems.
 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.

## Crudification of piecewise-quadratic representation.

o far we have run the cascade procedure ( Cascade algorithm ) according to the definition. On every step we produce an approximation while we intend for in where is a scaling function of finite support. Thus, we roughly double the number of pieces in a piecewise polynomial representation of on every step.

Therefore, we would like to periodically interrupt the cascade procedure to replace with some for some significant so that we would still have for small . Then we would restart the cascade procedure from .

Previously, see section ( Convergence of cascade procedure ), we experimented to establish optimality of piecewise quadratic representation of . Below we derive a recipe for the transformation in such case.

Let for some . Our intention is to use to approximate some function in . We drop the parameter from notation in this section: We use quadratic polynomials:

The cascade procedure preserves smoothness of the initial approximation. Hence, we assume that we have continuity of first derivative, see the section ( Convergence of cascade procedure ), and intend to preserve such property. Furthermore, we regard this situation in context of a large piecewise quadratic representation: we have a lot of pieces , we single out these four and aim to transform them. Thus, the boundary values at and come as input parameters. We arrive to the following requirements. where the are input parameters. We have 12 parameters and 10 conditions. At this point it should be apparent why we chose 4 intervals: this is the minimal even number of intervals that leaves out any freedom. We introduce two parameters to represent our degrees of freedom:
Piecewise Quadratic Polynomial picture

Next, we express the coefficients as functions of the input data and the control parameters . The following are the equations we are solving:

We will be repeatedly solving the following partial problem: where we want to express and as functions of other variables. We multiply the second equation with and add the two equations: Hence,
 (Partial solution)

We apply the recipe ( Partial solution ) to ( Quadratic polynomials 1 ) and ( Quadratic polynomials 3 ) and obtain We substitute these into ( Quadratic polynomials 2 ): or We simplify: We introduce the notation then or We add/subtract the equations and obtain or Consequently, We obtain resolution for the equations ( Quadratic polynomials 3 )-( Quadratic polynomials 5 ) by making the replacements Thus

We summarize our findings.

Proposition

(Quadratic piecewise interpolation) A piecewise polynomial of the form satisfies the conditions for given if and only if

Remark

Note that For even ,

Definition

(Crudification operator) Let be the class of piecewise quadratic functions with finite support defined on : We define a transformation according to the rule where the piecewise polynomial is constructed according to the following procedure.

Let For every let where and come from the proposition ( Quadratic piecewise interpolation ) with Then

We define for by recursion

Proposition

(Main property of crudification operator) Let then for some and for all .

Remark

In the code the rows for are pulled by a thread id and we interested in finding the corresponding rows for . For each thread we calculate two rows of . Thus, the correspondences are as follows were the is the row offset for . Thus