Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
3. Finite element method.
4. Construction of approximation spaces.
A. Finite element.
B. Averaged Taylor polynomial.
C. Stable space splittings.
D. Frames.
E. Tensor product splitting.
F. Sparse tensor product. Cure for curse of dimensionality.
a. Definition of sparse tensor product.
b. Wavelet estimates in Sobolev spaces.
c. Stability of wavelet splitting.
d. Stable splitting for tensor product of Sobolev spaces.
e. Approximation by sparse tensor product.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Sparse tensor product. Cure for curse of dimensionality.


e build on results of the section ( Elementary definitions of wavelet analysis ).

Condition

(Sparse tensor product setup) We introduce the mesh $\QTR{cal}{T}^{d}$ (see the section Wavelet analysis ) on MATH . We assume that MATH and MATH are dual GMRAs, scaling functions and wavelets on $\Delta$ , see the sections ( Compactly supported smooth biorthogonal wavelets ) and ( Construction of MRA and wavelets on half line or an interval ). We assume that $\phi$ satisfies condition ( Frame formula 1 ) of the proposition ( Frame property 2 ) and we showed how to achieve that in the section ( Spline functions ).

Note that we denote MATH the entire MATH -basis of semi-orthogonal functions constructed in the section ( Construction of MRA and wavelets on half line or an interval ). In particular, the statement of the proposition ( Reproduction of polynomials 4 ) adapted to GMRA would read MATH for some MATH , (thus, there are $n+1$ vanishing moments).

We keep the notation MATH but it now includes the space $V_{0}$ so that MATH is in the closure of the linear span of MATH with $W_{0}$ being what is $V_{0}$ in the notation of the definitions ( Generalized multiresolution analysis ) and ( Dual GMRA ).

Therefore, the index $d$ in the notation $\psi_{d,k}$ refers to the inclusion MATH rather than scale operation and the index $k$ refers to the numbering of functions produced by the procedure of the section ( Construction of MRA and wavelets on half line or an interval ) rather than shift operation. However, the old and new meaning of these indexes is close, in particular, there is $L^{2}$ -norm stability when changing these indexes, see the formulas ( Property of scale and transport 2 ),( Property of scale and transport 3 ).

We keep the notation $V_{d}$ for the spaces MATH with the property MATH

Moreover, MATH for some MATH .

We use the notation MATH which is finite because we restrict our attention to bounded sets. Furthermore, MATH

We will be extending wavelet approximation to the domain MATH .




a. Definition of sparse tensor product.
b. Wavelet estimates in Sobolev spaces.
c. Stability of wavelet splitting.
d. Stable splitting for tensor product of Sobolev spaces.
e. Approximation by sparse tensor product.

Notation. Index. Contents.


















Copyright 2007