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.

## Reduction to system of linear algebraic equations.

e start from the problem ( Backward discontinuous Galerkin time-discretization ): We seek such that for every and any , where is input data. By inclusion , the function has the form The functions are sought as linear combinations of a wavelet basis: for some finite index set . A system of linear equations is obtained by selecting for all and all .

We substitute definitions into and calculate

We treat the pairs , as a single indexes . The above equation has the form where is a column , is a matrix

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

Summary

(Reduction to system of linear algebraic equations summary) Let be a Hilbert space and the Hilbert space is densely and compactly embedded in : where the refers to the duality with respect to -topology. Let be a mapping measurable with respect to the time parameter . Let be a solution of the problem We seek an approximate solution in the form where is a basis in , is a finite selection of functions from , is a time interval: the is a selected integer and are numbers calculated for from the linear equations where we introduce double indexes , and The column is defined where comes from input data for : where the Proj is the projection in on the class of functions of the form . For the is the result of the prior step of calculation.

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

Summary

(Reduction to linear equation for q=1 summary) Let be a Hilbert space and the Hilbert space is densely and compactly embedded in : where the refers to the duality with respect to -topology. Let be a mapping measurable with respect to the time parameter . Let be a solution of the problem We seek an approximate solution in the form where is a basis in , is a finite selection of functions from , is a time interval: The numbers are calculated for from the linear equations where The column is defined Initial value comes from where Proj is orthogonal projection in on the set .