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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.
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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 $\eta_{m}$ MATH while we intend for MATH in $L_{2}$ where $\phi$ is a scaling function of finite support. Thus, we roughly double the number of pieces in a piecewise polynomial representation of $\eta_{m}$ on every step.

Therefore, we would like to periodically interrupt the cascade procedure to replace $\eta_{m}\in V_{m}$ with some MATH for some significant $d$ so that we would still have MATH for small $\varepsilon$ . Then we would restart the cascade procedure from $\eta_{m}^{d}$ .

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

Let MATH for some $d$ . Our intention is to use $p\left( x\right) $ to approximate some function $q\left( x\right) $ in MATH . We drop the parameter $d$ from notation in this section: MATH We use quadratic polynomials: MATH

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 $\Delta_{d,k}$ , we single out these four and aim to transform them. Thus, the boundary values at $x_{0}$ and $x_{4}$ come as input parameters. We arrive to the following requirements. MATH MATH where the MATH are input parameters. We have 12 parameters MATH 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 MATH to represent our degrees of freedom: MATH
Piecewise Quadratic Polynomial picture
Piecewise Quadratic Polynomial picture

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

MATH (Quadratic polynomials 1)
MATH (Quadratic polynomials 2)
MATH (Quadratic polynomials 3)
MATH (Quadratic polynomials 4)
MATH (Quadratic polynomials 5)
We will be repeatedly solving the following partial problem: MATH where we want to express $c$ and $b$ as functions of other variables. We multiply the second equation with $-x$ and add the two equations: MATH Hence,
MATH (Partial solution)

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

We summarize our findings.


(Quadratic piecewise interpolation) A piecewise polynomial $p\left( x\right) $ of the form MATH satisfies the conditions MATH for given MATH if and only if MATH MATH MATH MATH MATH MATH MATH MATH MATH MATH


Note that MATH For even $k$ , MATH


(Crudification operator) Let $\pi_{d}^{2,1}$ be the class of piecewise quadratic functions with finite support defined on MATH : MATH We define a transformation MATH according to the rule MATH where the piecewise polynomial MATH is constructed according to the following procedure.

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

We define MATH for MATH by recursion MATH


(Main property of crudification operator) Let MATH then MATH for some $q_{1},q_{2}$ and MATH for all $i\in\QTR{cal}{Z}$ .


In the code the rows for $\tilde{p}$ are pulled by a thread id $n$ and we interested in finding the corresponding rows for $p$ . For each thread $n$ we calculate two rows of $\tilde{p}$ . Thus, the correspondences are as follows MATH were the $m$ is the row offset for $p$ . Thus MATH

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Copyright 2007