I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 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.

## Solving N-dimensional PDEs.

he adaptive techniques of the previous chapters have one important flaw: the matrix of the problem is not sparse. If the problem is one dimensional then it is not important. However, as dimensionality increase, representation of solution inevitably grows in size. The prospect of keeping a non-sparse matrix puts restrictive limits on usefulness of the technique. We present a recipe for removing such limitation. The recipe is not yet tested.

Consider a 2-dimensional PDE problem solved over point basis in every dimension. For simplicity, we do not use sparse tensor product and evaluate it straight: We assume that the operator admits factorization thus We do not need to store a matrix . Instead, we only store two matrixes and . We find ourselves in the situation already described in the section ( Reduction to system of linear algebraic equations for Black PDE ).

We change the notation: It extends naturally to -dimensional situation. We remarked in the section ( Reduction to system of linear algebraic equations for Black PDE ) that the operation " " commutes with matrix multiplication. It does not, however, commute with addition. It leads to a peculiar form of matrix algebra where matrix multiplications can be calculated but matrix additions have to be kept preserved in form of a list.

Therefore, we cannot utilize inversion recipes of the section ( Summary for Black equation in case q=1 ) because these would lead to very long lists. Instead, we use preconditioners of the form where are solution matrixes for one-dimensional problems, obtained from the original -dimensional problem via some form of projection operation.

Once preconditioning is accomplished, we get a well conditioned matrix of the form That provides efficient recipe for evaluation of for any column . Thus, GMRES is applicable.

The described above technique applies if matrix of original problem has the property or if it has a form to start from. The technique has wide potential because multivariate Taylor expansion leads to the form .