(51) |

This is the symbol of the differential operator

(52) |

A preconditioned gradient descent method have the form

(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 . 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.