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.
C. Laplace quadrature.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Laplace quadrature.

e assume that the condition ( Generic parabolic PDE setup ) takes place. We further assume that the operator $A$ is $t$ -independent.

We extend the function $u$ to the entire $t$ -line MATH and apply the Laplace transform (see the section ( Laplace transform )) to the equation MATH and obtain MATH MATH Thus MATH where, according to the properties of $A$ listed in the condition ( Generic parabolic PDE setup ), the resolvent $R_{z}$ has poles on the positive side of the real axis. According to the section ( Laplace transform ) and for $0<\sigma<\alpha$ , MATH Then we want to transform the contour of integration MATH to give the term $e^{-tz}$ exponential decay of order $e^{-x^{2}}$ to allow for Gauss-Hermit quadrature, see the formula ( Gauss-Hermite Integration ). Such operation would impose analyticity requirements on $g$ and thus, decay requirements on $f$ and would depend on positioning of poles of $g$ . Let us assume that we can transform the contour into MATH without crossing any singularities of $g$ . Then we arrive to MATH and, after application of the formula ( Gauss-Hermite Integration ), MATH for some complex numbers MATH and elements MATH . Thus we reduced the original problem to solving several spacial elliptic problems MATH that may be executed in parallel.

Transformation of the contour may be unnecessary if the function $g\left( z\right) $ already have the exponential decay.

Notation. Index. Contents.

Copyright 2007