Preconditioners for Finite Dimensional Design Space

The previous section discussed the construction of preconditioners from their symbol in case the design space was a space of functions defined, for example, on the boundary of a domain. This is the infinite dimensional design space. In many applications one uses a fixed finite dimensional representation of the design space, using a set of prescribed shape functions. When the number of these functions is small one can use acceleration techniques such as BFGS. When that number grows and the number of BFGS steps required to solve the problem increases significantly, one may combine a preconditioner which is based on the infinite dimensional analysis.

We consider two design spaces. The first, ${\cal A}$, is a space of functions which is infinite dimensional and the second one, ${\cal A}_q$, is a subspace of the first and is represented in terms of $q$ functions, $f_j, j=1, \dots, q$. We also assume that the set $f_j, j=1, \dots, q$ is orthonormal with respect to the usual $L_2$ inner product on the boundary,

$$\int _{\partial\Omega} f_j(s) f_k(s) ds = \delta _{j,k},$$ (27)

where $\delta _{j,k}$ is the Kronecker delta. Functions in ${\cal A}_q$ are linear combination of the form $\sum _{j=1}^{d-1} \alpha _j f_j$, and the space ${\cal A}_q$ can be identified $I\!\!R^q$. We construct a mapping $P$ from ${\cal A}$ to $I\!\!R^q$ by,

$$(P \alpha) _j = \int _{\partial\Omega} \alpha (s) f_j(s) ds \qquad j=1, \dots , q.$$ (28)

Note that the transpose of the operator $P$ acts from the finite dimensional space $I\!\!R^q$ to ${\cal A}$ and is given by

$$(P^T \vec{\alpha})(s) = \sum _{j=1}^{d-1} \alpha _j f_j(s)$$ (29)

where $\vec{\alpha} = ( \alpha _1, \dots, \alpha _q)$.

Now we come to the point of relating gradients calculated with respect to the design space ${\cal A}$ to those calculated with respect to $I\!\!R^q$. Let

$$\delta J = \epsilon \int _{\partial \Omega} \tilde\alpha (s) g(s)... ...artial \Omega} ({\cal H} \tilde\alpha) (s) \tilde\alpha (s) ds + O(\epsilon ^3)$$ (30)

be the variation of the functional corresponding to a change in the design variables by $\epsilon \tilde \alpha$. The gradient with respect to ${\cal A}$ is certainly $\nabla J = g$, and the Hessian is ${\cal H}$. Now if the change in the design variables are done in the subspace ${\cal A}_q$, we consider $\tilde \alpha = \sum _{j=1}^q \tilde\alpha _j f_j$ and then a substitution into the above expression for $\delta J$ gives

$$\delta J = \epsilon \sum _{j=1}^q \tilde\alpha _j \int _{\partial... ...e \alpha _j\int _{\partial \Omega} ({\cal H} f_j)(s) f_k(s) ds + O(\epsilon ^3)$$ (31)

and in that case

$$\nabla J = \left( \begin{array}{c} g_1 \\ \vdots \\ g_q \end{array}\right) = \vec{g}$$ (32)

where $\vec{g} = (g_1, \dots, g_q)$, $g_j = \int _{\partial \Omega} f_j(s) g(s) ds$, and the Hessian for the subspace, ${\cal H}_q$, is related to the full Hessian ${\cal H}$ as

$$({\cal H}_q)_{j,k} = \int _{\partial \Omega} ({\cal H} f_j)(s) f_k(s) ds \qquad j,k=1, \dots ,q.$$ (33)

Notice that for the finite dimensional design space we can write,

$$\begin{array}{l} \tilde \alpha (s) = ( P^T \vec{\alpha}) (s) \\ \nabla J = P g \\ {\cal H}_q = P {\cal H} P^T. \end{array}$$ (34)

These are the abstract formulas for the discrete quantities for the subspace as a function of the same quantities on the infinite dimensional space.

A preconditioner for the finite dimensional design can be obtained by constructing first the infinite dimensional preconditioner and then using the above formula to get ${\cal R}_q = {\cal H}^{-1}_q$. The preconditioned iteration is

$$\vec{\alpha} \leftarrow \vec{\alpha} - \delta {\cal R}_q \vec g.$$ (35)

Example Consider the case ${\cal H} = -\frac{1}{\beta ^2} \frac{d^2}{dx^2}$ which appeared in one of the previous lectures. The finite dimensional preconditioner is constructed from the inverse of the finite dimensional Hessian

$$({\cal H}_q)_{j,k} = - \frac{1}{\beta ^2}\int _{\partial \Omega} f_j(s)\frac{d^2 f_k}{dx^2}(s) ds.$$ (36)