Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author

I. Wavelet calculations.
II. Calculation of approximation spaces in one dimension.
III. Calculation of approximation spaces in one dimension II.
IV. One dimensional problems.
1. Decomposition of payoff function in one dimension. Adaptive multiscaled approximation.
2. Constructing wavelet basis with Dirichlet boundary conditions.
3. Accelerated calculation of Gram matrix.
4. Adapting wavelet basis to arbitrary interval.
5. Solving one dimensional elliptic PDEs.
6. Discontinuous Galerkin technique II.
A. Rebalancing wavelet basis.
B. Reduction to system of linear algebraic equations.
7. Solving one dimensional Black PDE.
8. Solving one dimensional mean reverting equation.
V. Stochastic optimization in one dimension.
VI. Scalar product in N-dimensions.
VII. Wavelet transform of payoff function in N-dimensions.
VIII. Solving N-dimensional PDEs.
Downloads. Index. Contents.

Reduction to system of linear algebraic equations.


e start from the problem ( Backward discontinuous Galerkin time-discretization ): We seek $v\in S_{\tau}^{q}$ such that for every MATH and any MATH , MATH where $u_{T}$ is input data. By inclusion $v\in S_{\tau}^{q}$ , the function $v$ has the form MATH The functions MATH are sought as linear combinations of a wavelet basis: MATH for some finite index set $K_{n}$ . A system of linear equations is obtained by selecting MATH for all $r=0,...,q-1$ and all $s\in K_{n}$ .

We substitute definitions into MATH and calculate MATH

We treat the pairs MATH , MATH as a single indexes $i,j$ . The above equation has the form MATH where $c$ is a column MATH , MATH is a matrix

MATH (Matrix of discontinuous Galerkin)
and the column MATH comes from previous step or input data: MATH

Summary

(Reduction to system of linear algebraic equations summary) 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. Let MATH be a mapping MATH measurable with respect to the time parameter $t\in\lbrack0,T]$ . Let MATH be a solution of the problem MATH We seek an approximate solution MATH MATH in the form MATH where MATH is a basis in $H$ , $K_{n}$ is a finite selection of functions from MATH , $T_{n}$ is a time interval: MATH the $q$ is a selected integer and MATH are numbers calculated for $n=N-1,...,0$ from the linear equations MATH where we introduce double indexes MATH , MATH and MATH MATH The column MATH is defined MATH where $v_{n+1}$ comes from input data for $n=N-1$ : MATH where the Proj $_{H}$ is the projection in $H$ on the class of functions of the form MATH . For $n<N-1$ the $v_{n+1}$ is the result of the prior step of calculation.

We will discover in the following sections that we can frequently do successful calculations using $q=1$ . We adapt the formulas to such case: MATH We get MATH The column MATH is defined MATH For $n=N-1$ we have MATH For $n<N-1$ we use result of the previous step MATH In either case MATH

Summary

(Reduction to linear equation for q=1 summary) 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. Let MATH be a mapping MATH measurable with respect to the time parameter $t\in\lbrack0,T]$ . Let MATH be a solution of the problem MATH We seek an approximate solution MATH MATH in the form MATH where MATH is a basis in $H$ , $K_{n}$ is a finite selection of functions from MATH , $T_{n}$ is a time interval: MATH The numbers MATH are calculated for $n=N-1,...,0$ from the linear equations MATH where MATH MATH MATH The column MATH is defined MATH MATH MATH Initial value MATH comes from MATH where Proj MATH is orthogonal projection in $H$ on the set MATH .





Downloads. Index. Contents.


















Copyright 2007