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.
 1 Review of variational inequalities in maximization case.
 2 Penalized problem for mean reverting equation.
 3 Impossibility of backward induction.
 4 Stochastic optimization over wavelet basis.
 A. Choosing probing functions.
 B. Time discretization of penalty term.
 C. Implicit formulation of penalty term.
 D. Smooth version of penalty term.
 E. Solving equation with implicit penalty term.
 F. Removing stiffness from penalized equation.
 G. Mix of backward induction and penalty term approaches I.
 H. Mix of backward induction and penalty term approaches I. Implementation and results.
 I. Mix of backward induction and penalty term approaches II.
 J. Mix of backward induction and penalty term approaches II. Implementation and results.
 K. Review. How does it extend to multiple dimensions?
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Solving equation with implicit penalty term.

e intend to apply Newton technique to the equation ( Implicit equation with penalty term ): The equation has the form with We introduce the convenience notations and drop the upperscript :

We calculate the matrix of first derivatives: where is a Gram matrix is a diagonal matrix with small values and the matrix has spectral radius close to 1 (from above).

The Newton procedure is the iteration where we use We start Newton iterations from For small time step, the initial position is sufficiently close to the root of for convergence of Newton iterations.

Due to smallness of the function and boundedness of other components, the matrix is safely invertible for wide range of parameters.

Implementation of such approach is implemented in the script soNewton1.py, located in the directory OTSProjects/python/wavelets2. Experimentation shows that in order to keep the discrepancy to acceptable size, one has to take in the area of 1.e-3. If we do not decrease the time step with then the Newton procedure diverges. Such extreme stiffness is not a feature of the problem under consideration but a limitation of the Newton technique. We propose a more robust procedure in the following section.