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

Definition of sparse tensor product.


efinition

(Sparse tensor product) In context of the condition ( Sparse tensor product setup ), the sparse tensor product space is the space MATH See the section ( Tensor product of Hilbert spaces ) for the notation $\otimes$ .

In the span we take all possible integer sets MATH such that MATH and MATH and then for each MATH we take all possible integer sets MATH such that MATH . For each pair of combinations MATH and MATH we form a tensor product MATH and take the linear span of such tensor products.

We proceed to calculate dimensionality of $\hat{V}^{D}$ . In the following $a\cong b$ means MATH . The constant $C$ might change its meaning in process of calculation. For $N=2$ we have MATH MATH MATH We assume that for $N-1$ we have MATH and show that in $N$ dimensions we have MATH We calculate MATH Note that $2^{D}\equiv K$ is the size of the mesh $\QTR{cal}{T}^{D}$ , see the section ( Wavelet analysis ). Thus, MATH .

Proposition

(Dimension of tensor product) Let $K=2^{d}$ be the size of the one-dimensional mesh $\QTR{cal}{T}^{d}$ (see the context ( Sparse tensor product setup )) then MATH as $k\rightarrow\infty$ .





Notation. Index. Contents.


















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