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

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.
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.
Downloads. Index. Contents.

Adapting wavelet basis to arbitrary interval.


he file OTSProjects/python/wavelet2/dirichlet3a.py produces a multiscale basis for the interval MATH with Dirichlet boundary conditions. The basis has the following structure.

Let variable "basis" contain the result of the calculation. It is a dictionary, basis['xsi'] is a set of scaling functions MATH of scale $d_{0}=3$ . Exactly, $d_{0}$ is the minimal scale that satisfies the condition ( Sufficiently fine scale 2 ) and there are $2\cdot2^{d_{0}}$ functions marked by indexes from the set MATH . The parameter $d_{0}$ says how many functions correspond to an interval of unit length and there are two such intervals in MATH . The functions marked by indexes from $K_{d_{0},1}$ and $K_{d_{0},2}$ are boundary scaling functions for left and right boundaries. Because this construction is done for $n=5$ vanishing moments, there are 4 boundary scaling functions for each boundary: 5 functions because of connection of $n$ to length of support minus one boundary condition. Because of the condition ( Sufficiently fine scale 2 ), functions from $K_{d_{0},1}$ and $K_{d_{0},2}$ have non-overlapping supports and $K_{d_{0},I}$ is non-empty. The functions marked by $K_{d_{0},I}$ are internal functions equal to a scale-transport transformation of the same scaling function $\phi$ . All of these statements should be trivial after examining the section ( Adapting GMRA to interval [0,1] ).

The basis[3] contains scaling functions MATH that complement MATH to a basis of scale $d_{0}+1$ . Thus MATH also contains $2\cdot2^{d_{0}}$ functions marked by indexes from the set MATH . The index subsets MATH have similar meaning.

The variables basis[4],basis[5],... contain wavelet complements to the scales $d_{0}+2$ , $d_{0}+3$ ,...

If we compare the boundary functions MATH with MATH then we discover that there is one-to-one transformation between two set that consists of scaling and transport. Thus, if we want to adapt MATH to a different interval of the form MATH then all we have to do is place scaled functions MATH , MATH to the boundaries of $I_{d,k_{1},k_{2}}$ and fill in the rest by transformations of $\phi$ . From the structure of $I_{d,k_{a},k_{b}}$ we know how many functions we need. If the difference $k_{b}-k_{a}$ is too small to keep the condition ( Sufficiently fine scale 2 ) then we increase the scale. The only challenge is to find correct transport shifts.

We introduce the convenience notation MATH We assume that MATH Then we will have $2m$ boundary scaling functions on the interval $I_{d,k_{a},k_{b}}$ and $k_{b}-k_{a}-2m$ internal functions.

The function MATH is the left-most internal scaling function. Its support is centered within the interval MATH . We need to map it into a function with support centered within MATH . Thus we apply the transformation $S_{-d_{0}}^{\ast}$ (see the section ( Elementary definitions of wavelet analysis )) to place the center in MATH , shift it with $T_{k_{1}}^{\ast}$ for some $k_{1}$ to place the center into MATH and then scale it with $S_{d}^{\ast}$ into MATH . We chose the parameter $k_{1}$ to match MATH or MATH We perform the chain of transformations: MATH Therefore, we obtain left boundary functions for $I_{d,k_{a},k_{b}}$ by applying the same transformation to MATH .

We perform a similar argument on the right side. The function MATH (python notation) is the right-most internal scaling function. Its support is centered within the interval MATH . We need to map it into a function with support centered within MATH . Thus we apply the transformation $S_{-d_{0}}^{\ast}$ to place the center in MATH , shift it with $T_{k_{2}}^{\ast}$ for some $k_{2}$ to place the center into MATH and then scale it with $S_{d}^{\ast}$ into MATH . We chose the parameter $k_{2}$ to match MATH We perform the chain of transformations: MATH

Summary

(Adapting wavelet basis to arbitrary interval) Let MATH are scaling functions and wavelets of scale $d_{0}$ on the interval MATH with $n$ vanishing moments and Dirichlet boundary conditions. Let MATH We construct scaling function and wavelets MATH for interval MATH according to the following procedure: MATH MATH MATH

The functions MATH are constructed exactly the same way from MATH .

After we adapted basis to a given interval we might want to change scale: to go from MATH to MATH covering the same interval. The idea of transformation is the same.

Let MATH be a set of scaling functions on $I_{d,k_{a},k_{b}}$ . We would like to produce a set MATH for some scale $d_{1}$ . We noted before that $2^{d_{1}}$ $d_{1}$ -scaled functions cover a unit length. The length of the interval $I_{d,k_{a},k_{b}}$ is MATH . Thus MATH functions are needed to cover $I_{d,k_{a},k_{b}}$ at scale $d_{1}$ . The procedure works only if this is an integer. The sets MATH are calculated from the requirements MATH

The function MATH is centered within the interval MATH . We would like to map it to MATH centered within MATH . Thus, we apply the transformation MATH for some shift $k_{1}$ to be found from equating MATH We perform transformations MATH

Similarly, we perform calculation at the right side, equating support centers of MATH and MATH : MATH vs MATH . We transform MATH equate MATH and find MATH

Summary

(Changing scale of wavelet basis) Let MATH are scaling functions and wavelets of scale $d_{0}$ on the interval MATH with $n$ vanishing moments and Dirichlet boundary conditions. Let MATH We construct scaling function and wavelets MATH for the same interval according to the following procedure. Check that MATH MATH are integers. Then MATH MATH MATH

The functions MATH are constructed exactly the same way from MATH .

To enable the procedure of the section ( Accelerated calculation of Gram matrix ), we need to associate a fixed function to all MATH for all $d$ : MATH for some $\psi$ and a shift MATH . We find MATH by matching centers of supports. We fix MATH by requiring that $\psi$ would be centered at or around 0. We noted before that the function MATH is centered within the interval MATH where MATH . We perform the transformation MATH and require that the support would be around 0. MATH We need MATH If $2^{d-d_{0}}k_{a}$ is not an integer then we set MATH This way zero would be included in MATH for any $d,d_{0}$ .

Summary

(Common shape identification) Let MATH are scaling functions and wavelets of scale $d_{0}$ on the interval MATH with $n$ vanishing moments and Dirichlet boundary conditions. Let MATH The internal scaling functions and wavelets have the property MATH and some functions $\phi,\psi$ . In particular, MATH

Note that in all calculations of the section ( Accelerated calculation of Gram matrix ), applied to $\chi_{d,k}$ , $k$ has the meaning MATH .





Downloads. Index. Contents.


















Copyright 2007