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Quantitative Analysis
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Numerical Analysis
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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.

Discontinuous Galerkin technique.


ondition

(Generic parabolic PDE setup) Let $H$ be a Hilbert space and the Hilbert space $G$ is densely and compactly embedded in $H$ : MATH where the $G^{\ast}$ refers to the duality with respect to $H$ -topology. For example, MATH , MATH , compare with the chapter ( Parabolic PDE ). Let MATH be a mapping MATH measurable with respect to the time parameter $t\in\lbrack0,T]$ . For every $t$ the operator $A\left( t\right) $ is self adjoint and, uniformly in $t$ , has the properties MATH Let $u,f$ be mappings MATH We are considering the equation

MATH (Generic parabolic PDE problem)
for some $u_{0}\in G$ .




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.

Notation. Index. Contents.


















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