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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.

Ritz-Galerkin approximation.


et $X^{n}$ be an $n$ -dimensional subspace of the class MATH . We introduce the following problem.

Problem

(Approximate toy problem) For MATH find $u_{n}\in X^{n}$ such that MATH

We propose $u_{n}$ as an approximation of the solution $u$ of the problem ( Variational toy problem ). Since $u_{n}$ is a solution of a finite dimensional problem, its existence follows from uniqueness considerations and is easily proven.

Proposition

There exists a unique solution of the problem ( Approximate toy problem ).

Proof

Let MATH be a basis of $X^{n}$ . We seek the solution $u_{n}$ of the form MATH for a numerical sequence MATH . It suffices to have MATH The above is a linear system of algebraic equations for the definition of MATH with a square matrix. Hence, it suffices to prove that the system MATH can have only the trivial solution. To do so we perform integration by parts as in the proof of the proposition ( Equivalence of toy problems ) and arrive to the conclusion that $w$ must satisfy MATH From $w^{\prime\prime}=0$ and MATH we derive that $w^{\prime}=const$ . Consequently, from MATH and MATH we obtain $w=0$ .





Notation. Index. Contents.


















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