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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.
A. Tutorial introduction into finite element method.
a. Variational formulation, essential and natural boundary conditions.
b. Ritz-Galerkin approximation.
c. Convergence of approximate solution. Energy norm argument.
d. Approximation in L2 norm. Duality argument.
e. Example of finite dimensional subspace construction.
f. Adaptive approximation.
B. Finite elements for Poisson equation with Dirichlet boundary conditions.
C. Finite elements for Heat equation with Dirichlet boundary conditions.
D. Finite elements for Heat equation with Neumann boundary conditions.
E. Relaxed boundary conditions for approximation spaces.
F. Convergence of finite elements applied to nonsmooth data.
G. Convergence of finite elements for generic parabolic operator.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Example of finite dimensional subspace construction.

n this section we construct an example subspace $X^{n}$ and prove the condition ( Energy approximation ).

We introduce a "partition" set ("mesh") MATH MATH and define $X^{n}$ to be a class of functions $v$ such that MATH for all sets of numbers MATH and MATH .

We define MATH to be a set of functions from $X^{n}$ such that MATH The set MATH is a basis of $X^{n}$ . Indeed, it is linearly independent and for any $v\in X^{n}$ we have MATH Therefore, for a function MATH we define an approximation MATH and proceed to estimate MATH We introduce the convenience notations MATH We estimate MATH We make a linear change of variables that maps MATH into $\left[ 0,1\right] $ : MATH thus MATH We continue MATH and introduce the convenience notation MATH thus MATH Note that MATH by construction of $\tilde{u}$ . Hence, there is a MATH such that MATH then MATH we apply the formula ( Holder inequality ) MATH We substitute the last result into the estimate MATH We change the order of integration MATH We sum the estimate MATH for $i=2,...,n$ and obtain MATH

Notation. Index. Contents.

Copyright 2007