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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 is -independent.

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

Transformation of the contour may be unnecessary if the function already have the exponential decay.

 Notation. Index. Contents.
 Copyright 2007