Two Dimensional Case

In this case $k_2$ does not exists and we have a symbol

$$\hat{\cal H} (k_1) = \frac{k_1^2}{\vert 1- M^2 \vert }.$$ (51)

This is the symbol of the differential operator

$$- \frac{1}{\vert 1- M^2 \vert } \frac{\partial ^2 }{\partial x^2}.$$ (52)

A preconditioned gradient descent method have the form

$$\begin{array}{c} \mu \psi - \frac{1}{\vert 1 - M^2 }\vert \frac{d... ...al \lambda }{\partial x} \\ \alpha \leftarrow \alpha - \delta \psi \end{array}$$ (53)

That is, we have to solve an ODE on the boundary with the gradient as a source term, before using it as a direction of change for the design variable. According to our analysis the new method converges at a rate with is independent of the number of design variables, since the symbol for the modified iteration does not depend on ${\bf k}$. Note that the construction of the preconditioner was done on the differential level but the numerical implementation is using some approximation of it, e.g., finite difference approximation.