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.
 A. Tutorial introduction into finite element method.
 a. Variational formulation, essential and natural boundary conditions.
 b. Ritz-Galerkin approximation.
 c. Convergence of approximate solution. Energy norm argument.
 d. Approximation in L2 norm. Duality argument.
 e. Example of finite dimensional subspace construction.
 f. Adaptive approximation.
 B. Finite elements for Poisson equation with Dirichlet boundary conditions.
 C. Finite elements for Heat equation with Dirichlet boundary conditions.
 D. Finite elements for Heat equation with Neumann boundary conditions.
 E. Relaxed boundary conditions for approximation spaces.
 F. Convergence of finite elements applied to nonsmooth data.
 G. Convergence of finite elements for generic parabolic operator.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

Adaptive approximation.

et be a mesh on the interval , . We consider approximation of functions with a piecewise constant functions , and investigate the possibility to have an estimate of the form for some function .

First, we note that if we require that such estimate would hold for a fixed mesh and any then the function cannot have a desirable property Indeed, consider a mesh-dependent function : For the best piecewise constant approximation we have thus is not possible.

However, it is possible to have if the mesh is adapted to a given function . We start from and position to satisfy for some input parameter . We continue recursively Since we eventually stop at a step such that Let then Note that thus is connected to via a relationship of the form

 Notation. Index. Contents.
 Copyright 2007