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.

Adapting wavelet basis to arbitrary interval.

he file OTSProjects/python/wavelet2/dirichlet3a.py produces a multiscale basis for the interval 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 of scale . Exactly, is the minimal scale that satisfies the condition ( Sufficiently fine scale 2 ) and there are functions marked by indexes from the set . The parameter says how many functions correspond to an interval of unit length and there are two such intervals in . The functions marked by indexes from and are boundary scaling functions for left and right boundaries. Because this construction is done for vanishing moments, there are 4 boundary scaling functions for each boundary: 5 functions because of connection of to length of support minus one boundary condition. Because of the condition ( Sufficiently fine scale 2 ), functions from and have non-overlapping supports and is non-empty. The functions marked by are internal functions equal to a scale-transport transformation of the same scaling function . All of these statements should be trivial after examining the section ( Adapting GMRA to interval [0,1] ).

The basis[3] contains scaling functions that complement to a basis of scale . Thus also contains functions marked by indexes from the set . The index subsets have similar meaning.

The variables basis[4],basis[5],... contain wavelet complements to the scales , ,...

If we compare the boundary functions with 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 to a different interval of the form then all we have to do is place scaled functions , to the boundaries of and fill in the rest by transformations of . From the structure of we know how many functions we need. If the difference 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 We assume that Then we will have boundary scaling functions on the interval and internal functions.

The function is the left-most internal scaling function. Its support is centered within the interval . We need to map it into a function with support centered within . Thus we apply the transformation (see the section ( Elementary definitions of wavelet analysis )) to place the center in , shift it with for some to place the center into and then scale it with into . We chose the parameter to match or We perform the chain of transformations: Therefore, we obtain left boundary functions for by applying the same transformation to .

We perform a similar argument on the right side. The function (python notation) is the right-most internal scaling function. Its support is centered within the interval . We need to map it into a function with support centered within . Thus we apply the transformation to place the center in , shift it with for some to place the center into and then scale it with into . We chose the parameter to match We perform the chain of transformations:

Summary

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

The functions are constructed exactly the same way from .

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

Let be a set of scaling functions on . We would like to produce a set for some scale . We noted before that -scaled functions cover a unit length. The length of the interval is . Thus functions are needed to cover at scale . The procedure works only if this is an integer. The sets are calculated from the requirements

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

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

Summary

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

The functions are constructed exactly the same way from .

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

Summary

(Common shape identification) Let are scaling functions and wavelets of scale on the interval with vanishing moments and Dirichlet boundary conditions. Let The internal scaling functions and wavelets have the property and some functions . In particular,

Note that in all calculations of the section ( Accelerated calculation of Gram matrix ), applied to , has the meaning .