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.
4. Construction of approximation spaces.
5. Time discretization.
A. Change of variables for parabolic equation.
B. Discontinuous Galerkin technique.
a. Weak formulation with respect to time parameter.
b. Discretization with respect to time parameter.
c. Discretization for backward Kolmogorov equation.
d. Existence and uniqueness for time-discretized problem.
e. Convergence of discontinuous Galerkin technique. Adaptive time stepping.
C. Laplace quadrature.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Discretization with respect to time parameter.

n context of the condition ( Generic parabolic PDE setup ) we introduce partition of the interval $\left[ 0,T\right] $ : MATH We define MATH to be the class MATH We introduce the notations MATH for MATH .

We aim to approximate the solution $u$ of the problem ( Weak formulation with respect to time parameter ) with a function from $S_{\tau}^{q}$ for fixed $q$ and vanishing $\Delta t$ . We calculate for $v\in S_{\tau}^{q}$ and MATH : MATH Note that MATH We continue MATH The equation ( Time-weak problem ) transforms into MATH for any MATH . Thus, MATH . We replace MATH with $w\in S_{\tau}^{q}$ : MATH We can switch former problem for the latter if we can prove later that both problems have a unique solution. We merge conditions and arrive to MATH At this point we fix $T_{n}$ and take a $w$ with support within MATH . We arrive to the following problem.


(Discontinuous Galerkin time-discretization) In context of the condition ( Generic parabolic PDE setup ) we seek $v\in S_{\tau}^{q}$ such that for every MATH and any MATH , MATH

Note that the solution of the equation ( Generic parabolic PDE problem ) satisfies the above problem.

Notation. Index. Contents.

Copyright 2007