Optimization and Computational Fluid Dynamics - Department of ...

282 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen
the unit-process models and their outputs, we make an assumption that the
objectives are continuously differentiable or H-differentiable [7]. One should
note that in (9.7) the objectives fi,i=1,...,nf depend on the decision
variables and also, the unit-process model outputs. That is, the virtual papermaking
line model (9.6) has to be solved every time the objectives are
evaluated.
Next we discuss two features related to model-based optimization problems
such as (9.7). These features are gradient evaluations and reliability of the
used modeling approaches.
Optimization methods utilizing gradient information of the objectives
(gradient-based optimizers) have often been found computationally efficient
in engineering applications. However, gradient information needs to be calculated
when using these methods. The finite difference approaches are widely
used techniques for gradient evaluations, but a large number of decision variables
increases the number of the function evaluations making the approach
computationally time-consuming. Instead, more sophisticated techniques can
be used. We present briefly a technique based on chain rule approach and
implicit differentiation schema. For that we assume that objectives and unitprocess
models as well as model outputs are continuously differentiable or at
least H-differentiable [7] in nonsmooth cases. Then gradient of fi with respect
to the vector x is
∂fi(x,q 1,...,q nm)=∂xfi(x, q 1,...,q nm)+
nm�
∂q fi(x, q j 1,...,qnm)∂xq j(x,q 1,...,qj−1) j=1
(9.8)
where differentials ∂xfi(x,q 1 ,...,q nm )and∂q j fi(x, q 1 ,...,q nm ), for all i =
1,...,nf and j =1,...,nm are assumed to be known and
∂xq j(x, q 1,...,q j−1)=−∂q j Aj(x, q 1,...,q j) −1�
∂xAj(x,q 1,...,q j)+
�j−1
�
∂q Aj(x,q k 1 ,...,qj )∂xqk (x,q1 ,...,qk−1 ) , j =1,...,nm.
k=1
(9.9)
In this way, gradient calculations can be done model-wise and gradients of
objective functions can be put together after all the unit-processes have been
solved. Thus, different modeling approaches can be easily combined together
into the virtual papermaking line model. For more details on this technique,
we refer to [21].
The virtual papermaking line (9.6) consists of unit-process models from
different disciplines. When statistical or stochastic techniques are used, the
unit-process models are based on experimental data and on conditions prevailing
during data collection. That is, the models can give reliable results
only if there has been enough varying modeling data. Furthermore, ranges
of the modeling data should not be exceeded, which can cause problems