One approach to achieve this is to change the basis representing the design space such that
one is using orthonormal decomposition. For the shape design
problem it means that
we use a basis of shape functions where the representation of the
shape is

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and where the index stands for the multigrid level, and the changes of state and adjoint variables corresponding to changes in can be approximated on this level .

The idea is that these new shape functions, are going to be of increasing oscillations, due to their orthogonality. This implies that their effect on the solution will also be of increasing oscillations. Thus, we achieve a decomposition of the design space into subspaces of different smoothness as far as its effect on the solution. The implication of this is that a natural preconditioner of the optimization process is done. Since each level of those coarse grids on which optimization is to be done works independently of the other, at the relaxation step, we have removed the bad conditioning from the problem using this decomposition. The relaxation of the design variables is done now for the instead of the original . On level we update of the total design variables, .

`Relaxation: Level `

`(1) Relax State Equation `

`(2) Relax Adjoint Equation `

`(3) Update design variables`