Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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.
5. Time discretization.
A. Change of variables for parabolic equation.
a. Change of spacial variable for evolution equation.
b. Multiplicative change of unknown function for evolution equation.
c. Orthogonal transformation for evolution equation.
B. Discontinuous Galerkin technique.
C. Laplace quadrature.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Orthogonal transformation for evolution equation.

uppose we are facing a generic problem MATH where $u$ and $f$ are mappings MATH for a Hilbert space $H$ and MATH . Let $G$ be another Hilbert space and MATH be an orthogonal transformation MATH equivalently MATH Thus $G$ is any geometry-preserving transformation. In particular, change of basis fits.

We make the change of unknown function MATH Let MATH for some MATH . Then MATH

In context of parabolic PDE, $Q$ may be any transformation MATH that preserves $L^{2}$ -geometry. In particular, Fourier transform in $x$ -space fits as well as decomposition with respect to any orthonormal basis.

For example, let MATH be a $t$ -dependent basis of $H$ and MATH so that MATH or MATH We calculate MATH MATH MATH Thus, we remove $t$ -dependency from spacial operator if we can find an orthonormal basis MATH such that MATH

Notation. Index. Contents.

Copyright 2007