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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
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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.
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.
a. Elliptic problem with relaxed boundary approximation.
b. Parabolic problem with relaxed boundary approximation.
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.

Parabolic problem with relaxed boundary approximation.


roblem

(Partially inverted semi discrete parabolic problem) Assuming existence of the operator $T_{h}$ as in the condition ( Properties of solution operator ) we pose the problem of finding MATH such that MATH

Proposition

(Partial inversion lemma) Let MATH for $t\geq0$ where the operator $T_{h}$ is nonnegative MATH for some scalar product MATH . Then for the corresponding norm MATH we have MATH

Proof

We apply the operation MATH to the relationship $T_{h}e+e=\rho$ and obtain MATH We also have MATH and MATH . Hence, using nonnegativeness of $T_{h}$ MATH We integrate: MATH MATH Let MATH , MATH , then MATH It remains to note that MATH

Proposition

(Galerkin convergence 4) Assume that the condition ( Properties of solution operator ) is satisfied and let $u_{h}$ and $u$ be the solutions of the problems ( Heat equation with Dirichlet boundary condition ) and ( Partially inverted semi discrete parabolic problem ) respectively. We have MATH for $t\geq0$ .

Proof

We introduce $e=u_{h}-u$ and calculate MATH We want to get rid of all $u_{h}$ terms and all time derivatives. We substitute the equation MATH from the problem ( Partially inverted semi discrete parabolic problem ): MATH According to the definition ( Solution operator for elliptic problem ), we have $Tu_{t}+u=Tf$ : MATH According to the problem ( Heat equation with Dirichlet boundary condition ), we have $u_{t}-\Delta u=f$ : MATH We now apply the proposition ( Partial inversion lemma ) with MATH : MATH where MATH MATH By the condition ( Properties of solution operator ), MATH In exactly the same way MATH





Notation. Index. Contents.


















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