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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.
 a. Change of spacial variable for evolution equation.
 b. Multiplicative change of unknown function for evolution equation.
 c. Orthogonal transformation for evolution equation.
 B. Discontinuous Galerkin technique.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Orthogonal transformation for evolution equation.

uppose we are facing a generic problem where and are mappings for a Hilbert space and . Let be another Hilbert space and be an orthogonal transformation equivalently Thus is any geometry-preserving transformation. In particular, change of basis fits.

We make the change of unknown function Let for some . Then

In context of parabolic PDE, may be any transformation that preserves -geometry. In particular, Fourier transform in -space fits as well as decomposition with respect to any orthonormal basis.

For example, let be a -dependent basis of and so that or We calculate Thus, we remove -dependency from spacial operator if we can find an orthonormal basis such that

 Notation. Index. Contents.