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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.

Adaptive approximation.

et MATH be a mesh on the interval $\left[ 0,1\right] $ , MATH . We consider approximation of functions MATH with piecewise constant functions $\tilde{u}$ , MATH and investigate possibility to have an estimate of the form MATH for some function MATH .

First, we note that if we require that such estimate would hold for a fixed mesh and any MATH then the function $C\left( n\right) $ cannot have the desirable property MATH Indeed, consider a mesh-dependent function $u$ : MATH For the best piecewise constant approximation $\tilde{u}$ we have MATH thus MATH is not possible.

However, it is possible to have MATH if the mesh MATH is adapted to a given function $u$ . We start from $x_{0}=0$ and position $x_{1}$ to satisfy MATH for some input parameter $\Delta$ . We continue recursively MATH Since MATH we eventually stop at a step $n$ such that MATH Let MATH then MATH Note that MATH thus $\Delta$ is connected to $n$ via a relationship of the form MATH

Notation. Index. Contents.

Copyright 2007